5. Basic API tutorial


In this chapter the reader will learn how to build a simple application that uses MOSEK.

A number of examples is provided to demonstrate the functionality required for solving linear, quadratic, and conic problems as well as mixed integer problems.

Please note that the section on linear optimization also describes most of the basic functionality that is not specific to linear problems. Hence, it is recommended to read Section 5.2 before reading the rest of this chapter.

5.1. The basics

A typical program using the MOSEK Python interface can be described shortly:

  1. Create a handle to the MOSEK functionality.
  2. Create an environment (pymosek.Env) object.
  3. Set up some environment specific data and initialize the environment object.
  4. Create a task (pymosek.Task) object.
  5. Load a problem into the task object.
  6. Optimize the problem.
  7. Fetch the result.

5.1.1. The environment and the task

The first MOSEK related step in any program that employs MOSEK is to create an environment (pymosek.Env) object. The environment contains environment specific data such as information about the license file, streams for environment messages etc. Before creating any task objects, the environment must be initialized using Env.initenv. When this is done one or more task (pymosek.Task) objects can be created. Each task is associated with a single environment and defines a complete optimization problem as well as task message streams and optimization parameters.

In Python creation of an environment and a task would look something like this:

import pymosek
import Numeric

# Create a Mosek handle
msk = pymosek.mosek()
# Create an environment
env = msk.Env()

# You may connect streams and other callbacks to env here.

# Initialize the environment
env.init()
# Create a task
task = env.Task()

...
# Load a problem into the task, optimize etc.
...
# Fetch a solution from the task.

Please note that an environment should, if possible, be shared between multiple tasks.

5.1.2. A simple working example

The following simple example shows a working Python program which

  • creates an environment and a task,
  • reads a problem from a file,
  • optimizes the problem, and
  • writes the solution to a file.

# # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: simple.py # # Purpose: Demonstrates a very simple example using MOSEK by # reading a problem file, solving the problem and # writing the solution to a file. # import sys import pymosek as pymsk if len(sys.argv) <= 1: print "Missing argument. The syntax is:" print " simple inputfile [ solutionfile ]" else: # Make mosek handle msk = pymsk.mosek () # Make mosek environment. env = pymsk.Env () # Initialize the environment. env.init () # Create a task object linked with the environment env. # We create it initially with 0 variables and 0 columns, # since we don't know the size of the problem. task = env.Task (0,0) # We assume that a problem file was given as the first command # line argument (received in `args') task.readdata (sys.argv[1]) # Solve the problem task.optimize() # Print a summary of the solution task.solutionsummary(pymsk.streamtype.log) # If an output file was specified, write a solution if len(sys.argv) > 2: # We define the output format to be OPF, and tell MOSEK to # leave out parameters and problem data from the output file. task.putintparam (pymsk.iparam.write_data_format, pymsk.dataformat.op) task.putintparam (pymsk.iparam.opf_write_solutions, pymsk.onoffkey.on) task.putintparam (pymsk.iparam.opf_write_hints, pymsk.onoffkey.off) task.putintparam (pymsk.iparam.opf_write_parameters, pymsk.onoffkey.off) task.putintparam (pymsk.iparam.opf_write_problem, pymsk.onoffkey.off) task.writedata(sys.argv[2])

5.1.2.1. Writing a problem to a file

Use the Task.writedata function to write a problem to a file. By default MOSEK will determine the output file format by the extension of the filename, for example to write an OPF file:

   task.writedata("problem.opf");

5.1.2.2. Inputting and outputting problem data

An optimization problem consists of several components; objective, objective sense, constraints, variable bounds etc. Therefore, the task (pymosek.Task) provides a number of methods to operate on the task specific data, all of which are listed in Section 13.6.

5.1.2.3. Setting parameters

Apart from the problem data, the task contains a number of parameters defining the behavior of MOSEK. For example the mosek.iparam.optimizer parameter defines which optimizer to use. A complete list of all parameters are listed in Chapter 14.

5.1.3. Compiling and running examples

All examples presented in this chapter are distributed with MOSEK and are available in the directory

 mosek/5/tools/examples/ 

in the MOSEK installation. Chapter 4 describes how to compile and run the examples.

It is recommended to copy examples to a different directory before modifying and compiling them.

5.2. Linear optimization

The simplest optimization problem is a purely linear problem. A linear optimization problem is a problem of the following form:

Minimize or maximize the objective function

\begin{math}\nonumber{}\sum _{{j=0}}^{{n-1}}c_{j}x_{j}+c^{f}\end{math} (5.2.1)

subject to the linear constraints

\begin{math}\nonumber{}l_{k}^{c}\leq{}\sum _{{j=0}}^{{n-1}}a_{{kj}}x_{j}\leq{}u_{k}^{c},~k=0,\ldots ,m-1,\end{math} (5.2.2)

and the bounds

\begin{math}\nonumber{}l_{j}^{x}\leq{}x_{j}\leq{}u_{j}^{x},~j=0,\ldots ,n-1,\end{math} (5.2.3)

where we have used the problem elements

m and n, which are the number of constraints and variables respectively,

x, which is the variable vector of length n,

c, which is a coefficient vector of size n

\begin{displaymath}\nonumber{}c=\left[\begin{array}{c}\nonumber{}c_{0}\\\nonumber{}\vdots \\\nonumber{}c_{{n-1}}\end{array}\right],\end{displaymath}

[[MathCmd 5]], which is a scalar constant,

A, which is a [[MathCmd 6]] matrix of coefficients is given by

\begin{displaymath}\nonumber{}A=\left[\begin{array}{ccc}\nonumber{}a_{{0,0}} & \cdots  & a_{{0,(n-1)}}\\\nonumber{}\vdots  & \cdots  & \vdots \\\nonumber{}a_{{(m-1),0}} & \cdots  & a_{{(m-1),(n-1)}}\end{array}\right],\end{displaymath}

[[MathCmd 8]] and [[MathCmd 9]], which specify the lower and upper bounds on constraints respectively, and

[[MathCmd 10]] and [[MathCmd 11]], which specifies the lower and upper bounds on variables respectively.

Please note the unconventional notation using 0 as the first index rather than 1. Hence, [[MathCmd 12]] is the first element in variable vector x. This convention has been adapted from Python arrays which are indexed from 0.

5.2.1. Example: lo1

The following is an example of a linear optimization problem:

\begin{math}\nonumber{}\begin{array}{lccccccccl}\nonumber{}\mbox{maximize} & 3x_{0} & + & 1x_{1} & + & 5x_{2} & + & 1x_{3} &  & \\\nonumber{}\mbox{subject to} & 3x_{0} & + & 1x_{1} & + & 2x_{2} &  &  & = & 30,\\\nonumber{} & 2x_{0} & + & 1x_{1} & + & 3x_{2} & + & 1x_{3} & \geq{} & 15,\\\nonumber{} &  &  & 2x_{1} &  &  & + & 3x_{3} & \leq{} & 25,\end{array}\end{math} (5.2.4)

having the bounds

\begin{math}\nonumber{}\begin{array}{ccccc}\nonumber{}0 & \leq{} & x_{0} & \leq{} & \infty ,\\\nonumber{}0 & \leq{} & x_{1} & \leq{} & 10,\\\nonumber{}0 & \leq{} & x_{2} & \leq{} & \infty ,\\\nonumber{}0 & \leq{} & x_{3} & \leq{} & \infty .\end{array}\end{math} (5.2.5)

5.2.1.1. Source code

The data structures used in the following example will be explained in detail in 5.8.

The Python program included below, which solves this problem, is distributed with MOSEK and can be found in the directory

  mosek\5\tools\examp\ 

## # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: lo1.py # # Purpose: Demonstrates how to solve small linear # optimization problem using the MOSEK Python API. # ## import sys import os import pymosek as mosek try: from Numeric import array, Float, zeros, ones except ImportError: from mosekarr import array, Float, zeros, ones # Since the actual value of Infinity is ignores, we define it solely # for symbolic purposes: inf = 0.0 # Define a stream printer to grab output from MOSEK def streamprinter(text): sys.stdout.write(text) sys.stdout.flush() # We might write everything directly as a script, but it looks nicer # to create a function. def main (): # Open MOSEK and create an environment and task # Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task(0,0) # Attach a printer to the task task.set_Stream (mosek.streamtype.log, streamprinter) # Bound keys for constraints bkc = array ([mosek.boundkey.fx, mosek.boundkey.lo, mosek.boundkey.up]) # Bound values for constraints blc = array ([30.0, 15.0, -inf]) buc = array ([30.0, +inf, 25.0]) # Bound keys for variables bkx = array ([mosek.boundkey.lo, mosek.boundkey.ra, mosek.boundkey.lo, mosek.boundkey.lo]) # Bound values for variables blx = array ([ 0.0, 0.0, 0.0, 0.0]) bux = array ([+inf, 10.0, +inf, +inf]) # Objective coefficients c = array([ 3.0, 1.0, 5.0, 1.0 ]) # We input the A matrix column-wise # asub contains row indexes asub = array([ 0, 1, 0, 1, 2, 0, 1, 1, 2]) # acof contains coefficients acof = array([ 3.0, 2.0, 1.0, 1.0, 2.0, 2.0, 3.0, 1.0, 3.0 ]) # aptrb and aptre contains the offsets into asub and acof where # columns start and end respectively aptrb = array([ 0, 2, 5, 7 ]) aptre = array([ 2, 5, 7, 9 ]) numvar = len(bkx) numcon = len(bkc) # Load data into task print 'ptrb =',aptrb print 'ptre =',aptre print "numvar =",numvar print "numcon =",numcon task.inputdata (numcon,numvar, c, # linear objective coefficients 0.0, # objective fixed value aptrb, aptre, asub,acof, bkc,blc,buc, bkx,blx,bux) task.putobjsense(mosek.objsense.maximize) task.optimize () print "Solution summary" task.solutionsummary (mosek.streamtype.msg); xx = zeros(numvar, Float) print "point 13" task.getsolutionslice(mosek.soltype.bas, mosek.solitem.xx, 0, numvar, xx) print "point 14" print "x =", xx # call the main function try: main () except mosek.Exception, e: print "ERROR: %s" % str(e.errno) if e.msg is not None: print "\t%s" % e.msg sys.exit(1) except: import traceback traceback.print_exc() sys.exit(1)

5.2.1.2. Example code comments

The MOSEK environment:

Before setting up the optimization problem, a MOSEK environment must be created and initialized. This is done on the lines:

# Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init ()

We connect a call-back function to the environment log stream. In this case the call-back function simply prints messages to the standard output stream.

MOSEK optimization task:

Next, an empty task object is created:

# Create a task task = env.Task(0,0) # Attach a printer to the task task.set_Stream (mosek.streamtype.log, streamprinter)

We also connect a call-back function to the task log stream. Messages related to the task are passed to the call-back function. In this case the stream call-back function writes its messages to the standard output stream.

Inputting the problem data:

When the task has been created, data can be loaded into it. This happens here:

# Load data into task print 'ptrb =',aptrb print 'ptre =',aptre print "numvar =",numvar print "numcon =",numcon task.inputdata (numcon,numvar, c, # linear objective coefficients 0.0, # objective fixed value aptrb, aptre, asub,acof, bkc,blc,buc, bkx,blx,bux)

There are several different ways to set up an optimization problem; in this case we loaded the whole problem using a single function, Task.inputdata.

The ptrb, ptre, asub, and aval arguments define the constraint matrix A in the column ordered sparse format (for details, see Section 5.8.3.2).

The c argument is a full vector defining the objective function.

The precise relation between the arguments and the mathematical expressions in (5.2.1)[[MathCmd 15]](5.2.3) is as follows.

  • The linear terms in the constraints:

    \begin{math}\nonumber{}\begin{array}{rl}\nonumber{}a_{{\mathtt{sub[t],j}}}=\mathtt{val[t]}, & t=\mathtt{ptrb[j]},\ldots ,\mathtt{ptre[j]}-1,\\\nonumber{} & j=0,\ldots ,\mathtt{numvar}-1.\end{array}\end{math} (5.2.6)

    For an illustrated example of the meaning of ptrb and ptre see Section 5.8.3.2.

  • The linear terms in the objective:

    \begin{math}\nonumber{}c_{j}=\mathtt{c[j]},~j=0,\ldots ,\mathtt{numvar}-1\end{math} (5.2.7)
  • The bounds for the constraints are specified using the bkc, blc, and buc variables. The components of the bkc integer array specify the type of the bounds according to Table 5.1.

    Symbolic constant Lower bound Upper bound
    mosek.boundkey.fx finite identical to the lower bound
    mosek.boundkey.fr minus infinity plus infinity
    mosek.boundkey.lo finite plus infinity
    mosek.boundkey.ra finite finite
    mosek.boundkey.up minus infinity finite
    Table 5.1: Interpretation of the bound keys.

    For instance bkc[2]= mosek.boundkey.lo means that [[MathCmd 18]] and [[MathCmd 19]]. Finally, the numerical values of the bounds are given by

    \begin{math}\nonumber{}l_{k}^{c}=\mathtt{blc[k]},~k=0,\ldots ,\mathtt{numcon}-1\end{math} (5.2.8)

    and

    \begin{math}\nonumber{}u_{k}^{c}=\mathtt{buc[k]},~k=0,\ldots ,\mathtt{numcon}-1.\end{math} (5.2.9)
  • The bounds on the variables are specified using the bkx, blx, and bux variables. The components in the bkx integer array specifies the type of the bounds according to Table 5.1. The numerical values for the lower bounds on the variables are given by

    \begin{math}\nonumber{}l_{j}^{x}=\mathtt{blx[j]},~j=0,\ldots ,\mathtt{numvar}-1.\end{math} (5.2.10)

    The numerical values for the upper bounds on the variables are given by

    \begin{math}\nonumber{}u_{j}^{x}=\mathtt{bux[j]},~j=0,\ldots ,\mathtt{numvar}-1.\end{math} (5.2.11)
Optimization:

After set-up the task can be optimized.

task.optimize ()
Outputting the solution:

Finally, the primal solution is retrieved and printed.

xx = zeros(numvar, Float) print "point 13" task.getsolutionslice(mosek.soltype.bas, mosek.solitem.xx, 0, numvar, xx) print "point 14" print "x =", xx

The Task.getsolutionslice function obtains a “slice” of the solution. In fact MOSEK may compute several solutions depending on the optimizer employed. In this example the basic solution is requested, specified by mosek.soltype.bas. The mosek.solitem.xx specifies that we want the variable values of the solution, and the following 0 and NUMVAR specifies the range of variable values we want.

The range specified is the first index (here “0”) up to but not including the second index (here “NUMVAR).

Catching exceptions:

We cache any exceptions thrown by mosek in the lines:

except mosek.Exception, e: print "ERROR: %s" % str(e.errno) if e.msg is not None: print "\t%s" % e.msg sys.exit(1)

The types of exceptions that MOSEK can throw can be seen in 13.4 and 13.7.

5.2.2. An alternative implementation: lo2

In the previous example the problem data is loaded in one chunk. It is often more convenient to add one constraint or one variable at a time — this is possible using the following approach:

  • Before a constraint or a variable can be used it has to be added with Task.append or a similar function. By default the appended constraints will be empty and the bounds of the appended constraints are infinite. Variables are fixed at zero.
  • The objective function is specified using Task.putcfix and Task.putcj.
  • The lower and upper bounds on the constraints and variables are specified using Task.putbound.
  • The non-zero entries in A are added one column at a time using Task.putavec.

# # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: lo2.py # # Purpose: Demonstrates how to solve small linear # optimization problem using the MOSEK Python API. ## import sys import pymosek as mosek from mosekarr import array, Float, zeros, ones # Since the actual value of Infinity is ignores, we define it solely # for symbolic purposes: inf = 0.0 # Define a stream printer to grab output from MOSEK def streamprinter(text): sys.stdout.write(text) sys.stdout.flush() # We might write everything directly as a script, but it looks nicer # to create a function. def main (): # Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task(0,0) # Attach a printer to the task task.set_Stream (mosek.streamtype.log, streamprinter) # Bound keys for constraints bkc = array ([mosek.boundkey.fx, mosek.boundkey.lo, mosek.boundkey.up]) # Bound values for constraints blc = array ([30.0, 15.0, -inf]) buc = array ([30.0, +inf, 25.0]) # Bound keys for variables bkx = array ([mosek.boundkey.lo, mosek.boundkey.ra, mosek.boundkey.lo, mosek.boundkey.lo]) # Bound values for variables blx = array ([ 0.0, 0.0, 0.0, 0.0]) bux = array ([+inf, 10.0, +inf, +inf]) # Objective coefficients csub = array([ 0, 1, 2, 3 ]) cval = array([ 3.0, 1.0, 5.0, 1.0 ]) # We input the A matrix column-wise # asub contains row indexes asub = array([ 0, 1, 0, 1, 2, 0, 1, 1, 2]) # acof contains coefficients acof = array([ 3.0, 2.0, 1.0, 1.0, 2.0, 2.0, 3.0, 1.0, 3.0 ]) # aptrb and aptre contains the offsets into asub and acof where # columns start and end respectively aptrb = array([ 0, 2, 5, 7 ]) aptre = array([ 2, 5, 7, 9 ]) numvar = len(bkx) numcon = len(bkc) # Append the constraints task.append(mosek.accmode.con,numcon); # Append the variables. task.append(mosek.accmode.var,numvar); # Input objective task.putcfix(0.0) task.putclist(csub,cval); # Put constraint bounds task.putboundslice(mosek.accmode.con, 0, numcon, bkc, blc, buc); # Put variable bounds task.putboundslice(mosek.accmode.var, 0, numvar, bkx, blx, bux); # Input A non-zeros by columns for j in range(numvar): ptrb,ptre = aptrb[j],aptre[j] task.putavec(mosek.accmode.var,j, asub[ptrb:ptre], acof[ptrb:ptre]) # Input the objective sense (minimize/maximize) task.putobjsense(mosek.objsense.maximize) # Optimize the task task.optimize(); # Output a solution xx = zeros(numvar, Float) task.getsolutionslice(mosek.soltype.bas, mosek.solitem.xx, 0,numvar, xx) print "xx =", xx # call the main function try: main () except mosek.Exception, (code,msg): print "ERROR: %s" % str(code) if msg is not None: print "\t%s" % msg sys.exit(1) except: import traceback traceback.print_exc() sys.exit(1)

5.3. Quadratic optimization

It is possible to solve quadratic and quadratically constrained convex problems using MOSEK. This class of problems can be formulated as follows:

\begin{math}\nonumber{}\begin{array}{lrcccll}\nonumber{}\mbox{minimize} &  &  & \frac{1}{2}x^{T}Q^{o}x+c^{T}x+c^{f} &  &  & \\\nonumber{}\mbox{subject to} & l_{k}^{c} & \leq{} & \frac{1}{2}x^{T}Q^{k}x+\sum \limits _{{j=0}}^{{n-1}}a_{{k,j}}x_{j} & \leq{} & u_{k}^{c}, & k=0,\ldots ,m-1,\\\nonumber{} & l^{x} & \leq{} & x & \leq{} & u^{x}, & j=0,\ldots ,n-1.\end{array}\end{math} (5.3.1)

Without loss of generality it is assumed that [[MathCmd 25]] and [[MathCmd 26]] are all symmetric because

\begin{displaymath}\nonumber{}x^{T}Qx=0.5x^{T}(Q+Q^{T})x.\end{displaymath}

This implies that a non-symmetric Q can be replaced by the symmetric matrix [[MathCmd 28]].

A very important restriction in MOSEK is that the problem should be convex. This implies that the matrix [[MathCmd 25]] should be positive semi-definite and that the kth constraint must be of the form

\begin{math}\nonumber{}l_{k}^{c}\leq{}\frac{1}{2}x^{T}Q^{k}x+\sum \limits _{{j=0}}^{{n-1}}a_{{k,j}}x_{j}\end{math} (5.3.2)

with a negative semi-definite [[MathCmd 26]], or of the form

\begin{math}\nonumber{}\frac{1}{2}x^{T}Q^{k}x+\sum \limits _{{j=0}}^{{n-1}}a_{{k,j}}x_{j}\leq{}u_{k}^{c}.\end{math} (5.3.3)

with a positive semi-definite [[MathCmd 26]]. This implies that quadratic equalities are specifically not allowed.

5.3.1. Example: Quadratic objective

The following is an example if a quadratic, linearly constrained problem:

\begin{math}\nonumber{}\begin{array}{lcccl}\nonumber{}\mbox{minimize} &  &  & x_{1}^{2}+0.1x_{2}^{2}+x_{3}^{2}-x_{1}x_{3}-x_{2} & \\\nonumber{}\mbox{subject to} & 1 & \leq{} & x_{1}+x_{2}+x_{3} & \\\nonumber{} &  &  & x\geq{}0 &\end{array}\end{math} (5.3.4)

This can be written equivalently as

\begin{math}\nonumber{}\begin{array}{lccl}\nonumber{}\mbox{minimize} & 1/2x^{T}Q^{o}x+c^{T}x &  & \\\nonumber{}\mbox{subject to} & Ax & \geq{} & b\\\nonumber{} & x & \geq{} & 0,\end{array}\end{math} (5.3.5)

where

\begin{math}\nonumber{}Q^{o}=\left[\begin{array}{ccc}\nonumber{}2 & 0 & -1\\\nonumber{}0 & 0.2 & 0\\\nonumber{}-1 & 0 & 2\end{array}\right],\quad{}c=\left[\begin{array}{c}\nonumber{}0\\\nonumber{}-1\\\nonumber{}0\end{array}\right],\quad{}A=\left[\begin{array}{ccc}\nonumber{}1 & 1 & 1\end{array}\right],\mbox{ and }b=1.\end{math} (5.3.6)

Please note that MOSEK always assumes that there is a 1/2 in front of the [[MathCmd 37]] term in the objective. Therefore, the 1 in front of [[MathCmd 38]] becomes 2 in Q, i.e. [[MathCmd 39]].

5.3.1.1. Source code

## # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: qo1.py # # Purpose: Demonstrate how to solve a quadratic # optimization problem using the MOSEK Python API. ## import sys import os import pymosek as mosek from mosekarr import array, Float, zeros, ones # Since the actual value of Infinity is ignores, we define it solely # for symbolic purposes: inf = 0.0 # Define a stream printer to grab output from MOSEK def streamprinter(text): sys.stdout.write(text) sys.stdout.flush() # We might write everything directly as a script, but it looks nicer # to create a function. def main (): # Open MOSEK and create an environment and task # Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task() # Set up and input bounds and linear coefficients bkc = array([ mosek.boundkey.lo ]) blc = array([ 1.0 ]) buc = array([ inf ]) bkx = array([ mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo ]) blx = array([ 0.0, 0.0, 0.0 ]) bux = array([ inf, inf, inf ]) c = array([ 0.0, -1.0, 0.0 ]) asub = array([ 0, 0, 0 ]) aval = array([ 1.0, 1.0, 1.0 ]) aptrb = array([ 0, 1, 2 ]) aptre = array([ 1, 2, 3 ]) numvar = len(bkx) numcon = len(bkc) numanz = len(asub) task.inputdata(numcon,numvar, c,0.0, aptrb, aptre, asub, aval, bkc, blc, buc, bkx, blx, bux) # Set up and input quadratic objective qsubi = array([ 0, 1, 2, 2 ]) qsubj = array([ 0, 1, 0, 2 ]) qval = array([ 2.0, 0.2, -1.0, 2.0 ]) task.putqobj(qsubi,qsubj,qval) task.putobjsense(mosek.objsense.minimize) # Optimize task.optimize() # Fetch the interior solution xx = zeros(numvar, Float) task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0, numvar, xx); print "Primal interior solution: " print "\t",xx return None # call the main function try: main() except mosek.Exception, e: print "ERROR: %s" % str(e.errno) if e.msg is not None: import traceback traceback.print_exc() print "\t%s" % e.msg sys.exit(1) except: import traceback traceback.print_exc() sys.exit(1) print "Finished OK" sys.exit(0)

5.3.1.2. Example code comments

Most of the functionality in this example has already been explained for the linear optimization example in Section 5.2 and it will not be repeated here.

This example introduces one new function, Task.putqobj, which is used to input the quadratic terms of the objective function.

Since [[MathCmd 25]] is symmetric only the lower triangular part of [[MathCmd 25]] is inputted. The upper part of [[MathCmd 25]] is computed by MOSEK using the relation

\begin{displaymath}\nonumber{}Q^{o}_{{ij}}=Q^{o}_{{ji}}.\end{displaymath}

Entries from the upper part may not appear in the input.

The lower triangular part of the matrix [[MathCmd 25]] is specified using an unordered sparse triplet format (for details, see Section 5.8.3):

qsubi = array([ 0, 1, 2, 2 ]) qsubj = array([ 0, 1, 0, 2 ]) qval = array([ 2.0, 0.2, -1.0, 2.0 ])

Please note that

  • only non-zero elements are specified (any element not specified is 0 by definition),
  • the order of the non-zero elements is insignificant, and
  • only the lower triangular part should be specified.

Finally, the matrix [[MathCmd 25]] is loaded into the task:

task.putqobj(qsubi,qsubj,qval)

5.3.2. Example: Quadratic constraints

In this section describes how to solve a problem with quadratic constraints. Please note that quadratic constraints are subject to the convexity requirement (5.3.2).

Consider the problem:

\begin{math}\nonumber{}\begin{array}{lcccl}\nonumber{}\mbox{minimize} &  &  & x_{1}^{2}+0.1x_{2}^{2}+x_{3}^{2}-x_{1}x_{3}-x_{2} & \\\nonumber{}\mbox{subject to} & 1 & \leq{} & x_{1}+x_{2}+x_{3}-x_{1}^{2}-x_{2}^{2}-0.1x_{3}^{2}+0.2x_{1}x_{3}, & \\\nonumber{} &  &  & x\geq{}0. &\end{array}\end{math} (5.3.7)

This is equivalent to

\begin{math}\nonumber{}\begin{array}{lccl}\nonumber{}\mbox{minimize} & 1/2x^{T}Q^{o}x+c^{T}x &  & \\\nonumber{}\mbox{subject to} & 1/2x^{T}Q^{0}x+Ax & \geq{} & b,\end{array}\end{math} (5.3.8)

where

\begin{math}\nonumber{}Q^{o}=\left[\begin{array}{ccc}\nonumber{}2 & 0 & -1\\\nonumber{}0 & 0.2 & 0\\\nonumber{}-1 & 0 & 2\end{array}\right],\quad{}c=\left[\begin{array}{c}\nonumber{}0\\\nonumber{}-1\\\nonumber{}0\end{array}\right],\quad{}A=\left[\begin{array}{ccc}\nonumber{}1 & 1 & 1\end{array}\right],\quad{}b=1.\end{math} (5.3.9)
\begin{math}\nonumber{}Q^{0}=\left[\begin{array}{ccc}\nonumber{}-2 & 0 & 0.2\\\nonumber{}0 & -2 & 0\\\nonumber{}0.2 & 0 & -0.2\end{array}\right].\end{math} (5.3.10)

5.3.2.1. Source code

## # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: qcqo1.py # # Purpose: Demonstrate how to solve a quadratic # optimization problem using the MOSEK API. # # minimize x0^2 + 0.1 x1^2 + x2^2 - x0 x2 - x1 # s.t 1 <= x0 + x1 + x2 - x0^2 - x1^2 - 0.1 x2^2 + 0.2 x0 x2 # x >= 0 ## import sys import os import pymosek as mosek from mosekarr import array, Float,zeros,ones # Since the actual value of Infinity is ignores, we define it solely # for symbolic purposes: inf = 0.0 # Define a stream printer to grab output from MOSEK def streamprinter(text): sys.stdout.write(text) sys.stdout.flush() # We might write everything directly as a script, but it looks nicer # to create a function. def main (): # Open MOSEK and create an environment and task # Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task() # Set up and input bounds and linear coefficients bkc = array([ mosek.boundkey.lo ]) blc = array([ 1.0 ]) buc = array([ inf ]) bkx = array([ mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo ]) blx = array([ 0.0, 0.0, 0.0 ]) bux = array([ inf, inf, inf ]) c = array([ 0.0, -1.0, 0.0 ]) asub = array([ 0, 0, 0 ]) aval = array([ 1.0, 1.0, 1.0 ]) aptrb = array([ 0, 1, 2 ]) aptre = array([ 1, 2, 3 ]) numvar = len(bkx) numcon = len(bkc) task.inputdata(numcon,numvar, c,0.0, aptrb, aptre, asub, aval, bkc, blc, buc, bkx, blx, bux) # Set up and input quadratic objective qsubi = array([ 0, 1, 2, 2 ]) qsubj = array([ 0, 1, 0, 2 ]) qval = array([ 2.0, 0.2, -1.0, 2.0 ]) task.putqobj(qsubi,qsubj,qval) # The lower triangular part of the Q^0 # matrix in the first constraint is specified. # This corresponds to adding the term # - x0^2 - x1^2 - 0.1 x2^2 + 0.2 x0 x2 qsubi = array([ 0, 1, 2, 2 ]) qsubj = array([ 0, 1, 2, 0 ]) qval = array([ -2.0, -2.0, -0.2, 0.2 ]) # put Q^0 in constraint with index 0. task.putqconk (0, qsubi,qsubj, qval); task.putobjsense(mosek.objsense.minimize) # Optimize task.optimize() # Fetch the interior solution xx = zeros(numvar,Float) task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0,numvar, xx) print "Primal interior solution: " print "\t",xx return None # call the main function try: main () except mosek.Exception, err: print "ERROR: %s" % str(err) sys.exit(1) except: import traceback traceback.print_exc() sys.exit(1)

The only new function introduced in this example is Task.putqconk, which is used to add quadratic terms to the constraints. While Task.putqconk add quadratic terms to a specific constraint, it is also possible to input all quadratic terms in all constraints in one chunk using the Task.putqcon function.

5.4. Conic optimization

Conic problems are a generalization of linear problems, allowing constraints of the type

\begin{displaymath}\nonumber{}x\in{}\mathcal{C}\end{displaymath}

where [[MathCmd 51]] is a convex cone.

MOSEK can solve conic optimization problems of the following form

\begin{math}\nonumber{}\begin{array}{lccccl}\nonumber{}\mbox{minimize} &  &  & c^{T}x+c^{f} &  & \\\nonumber{}\mbox{subject to} & l^{c} & \leq{} & Ax & \leq{} & u^{c},\\\nonumber{} & l^{x} & \leq{} & x & \leq{} & u^{x},\\\nonumber{} &  &  & x\in{}\mathcal{C} &  &\end{array}\end{math} (5.4.1)

where [[MathCmd 51]] is a cone. [[MathCmd 51]] can be a product of cones, i.e.

\begin{displaymath}\nonumber{}\mathcal{C}=\mathcal{C}_{0}\times \cdots \times \mathcal{C}_{{p-1}}\end{displaymath}

in which case [[MathCmd 56]] means [[MathCmd 57]]. Please note that the set of real numbers [[MathCmd 58]] is itself a cone, so linear variables are still allowed.

MOSEK supports two specific cones apart from the real numbers:

When creating a conic problem in MOSEK, each cone is defined by a cone type (quadratic or rotated quadratic cone) and a list of variable indexes. To summarize:

5.4.1. Example: cqo1

The problem

\begin{math}\nonumber{}\begin{array}{lccccc}\nonumber{}\mbox{minimize} & x_{4}+x_{5} &  & \\\nonumber{}\mbox{subject to} & x_{0}+x_{1}+x_{2}+x_{3} & = & 1,\\\nonumber{} & x_{0},x_{1},x_{2},x_{3} & \geq{} & 0,\\\nonumber{} & x_{4}\geq{}\sqrt{x_{0}^{2} + x_{2}^{2}}, &  & \\\nonumber{} & x_{5}\geq{}\sqrt{x_{1}^{2} + x_{3}^{2}} &  &\end{array}\end{math} (5.4.2)

is an example of a conic quadratic optimization problem. The problem includes a set of linear constraints and two quadratic cones.

5.4.1.1. Source code

# # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: cqo1.py # # Purpose: Demonstrates how to solve a small conic qaudratic # optimization problem using the MOSEK API. # import sys import os import pymosek as mosek try: from Numeric import array, Float, zeros,ones except ImportError: from mosekarr import array, Float, zeros,ones # Since the actual value of Infinity is ignores, we define it solely # for symbolic purposes: inf = 0.0 # Define a stream printer to grab output from MOSEK def streamprinter(text): sys.stdout.write(text) sys.stdout.flush() # We might write everything directly as a script, but it looks nicer # to create a function. def main (): # Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task(0,0) # Attach a printer to the task task.set_Stream (mosek.streamtype.log, streamprinter) bkc = array([ mosek.boundkey.fx ]) blc = array([ 1.0 ]) buc = array([ 1.0 ]) c = array([ 0.0, 0.0, 0.0, 0.0, 1.0, 1.0 ]) bkx = array([ mosek.boundkey.lo,mosek.boundkey.lo,mosek.boundkey.lo, mosek.boundkey.lo,mosek.boundkey.fr,mosek.boundkey.fr ]) blx = array([ 0.0, 0.0, 0.0, 0.0, -inf, -inf ]) bux = array([ inf, inf, inf, inf, inf, inf ]) asub = array([ 0, 0, 0, 0 ]) acof = array([ 1.0, 1.0, 1.0, 1.0 ]) # column 0 1 2 3 4 5 ptrb = array([ 0, 1, 2, 3, 4, 4 ]) ptre = array([ 1, 2, 3, 4, 4, 4 ]) numvar = len(bkx) numcon = len(bkc) # Load data into task task.inputdata (numcon,numvar, c, # linear objective coefficients 0.0, # objective fixed value ptrb, ptre, asub,acof, bkc,blc,buc, bkx,blx,bux) # Input the cones task.appendcone(mosek.conetype.quad, 0.0, array([ 4, 0, 2 ])) task.appendcone(mosek.conetype.quad, 0.0, array([ 5, 1, 3 ])) # Input objective sense task.putobjsense(mosek.objsense.minimize) # Optimize task.optimize () # Output the solution task.solutionsummary (mosek.streamtype.msg); xx = zeros(numvar, Float) task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0, numvar, xx) print "x =", xx return None # call the main function try: main () except mosek.Exception, e: print "ERROR: %s" % str(e.errno) if e.msg is not None: print "\t%s" % e.msg sys.exit(1) except: import traceback traceback.print_exc() sys.exit(1)

5.4.1.2. Source code comments

The only new function introduced in the example is Task.appendcone, which is called here:

task.appendcone(mosek.conetype.quad, 0.0, array([ 4, 0, 2 ]))

Here mosek.conetype.quad defines the cone type, in this case it is a quadratic cone. The cone parameter 0.0 is currently not used by MOSEK — simply passing 0.0 will work. c c

5.5. Integer optimization

An optimization problem where one or more of the variables are constrained to integer values is denoted an integer optimization problem.

5.5.1. Example: milo1

In this section the example

\begin{math}\nonumber{}\begin{array}{lccl}\nonumber{}\mbox{maximize} & x_{0}+0.64x_{1} &  & \\\nonumber{}\mbox{subject to} & 50x_{0}+31x_{1} & \leq{} & 250,\\\nonumber{} & 3x_{0}-2x_{1} & \geq{} & -4,\\\nonumber{} & x_{0},x_{1}\geq{}0 &  & \mbox{and integer}\end{array}\end{math} (5.5.1)

is used to demonstrate how to solve a problem with integer variables.

5.5.1.1. Source code

The example (5.5.1) is almost identical to a linear optimization problem except for some variables being integer constrained. Therefore, only the specification of the integer constraints requires something new compared to the linear optimization problem discussed previously. In MOSEK these constraints are specified using the function Task.putvartype as shown in the code:

task.putvartypelist(array([ 0, 1 ]), array([ mosek.variabletype.type_int, mosek.variabletype.type_int ]))

The complete source for the example is listed below.

## # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: mio1.cs # # Purpose: Demonstrates how to solve a small mixed # integer linear optimization problem using the MOSEK Python API. ## import sys import pymosek as mosek from mosekarr import array, Float, zeros, ones # Since the actual value of Infinity is ignores, we define it solely # for symbolic purposes: inf = 0.0 # Define a stream printer to grab output from MOSEK def streamprinter(text): sys.stdout.write(text) sys.stdout.flush() # We might write everything directly as a script, but it looks nicer # to create a function. def main (): # Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task(0,0) # Attach a printer to the task task.set_Stream (mosek.streamtype.log, streamprinter) bkc = array([ mosek.boundkey.up, mosek.boundkey.lo ]) blc = array([ -inf, -4.0 ]) buc = array([ 250.0, inf ]) bkx = array([ mosek.boundkey.lo, mosek.boundkey.lo ]) blx = array([ 0.0, 0.0 ]) bux = array([ inf, inf ]) c = array([ 1.0, 0.64 ]) asub = array([ 0, 1, 0, 1 ]) acof = array([ 50.0, 3.0, 31.0, -2.0 ]) ptrb = array([ 0, 2 ]) ptre = array([ 2, 4 ]) numvar = len(bkx) numcon = len(bkc) # Input linear data task.inputdata(numcon,numvar, c,0.0, ptrb, ptre, asub, acof, bkc, blc, buc, bkx, blx, bux) # Input objective sense task.putobjsense(mosek.objsense.maximize) # Define variables to be integers task.putvartypelist(array([ 0, 1 ]), array([ mosek.variabletype.type_int, mosek.variabletype.type_int ])) task.putintparam (mosek.iparam.mio_mode, mosek.miomode.satisfied) # Optimize task.optimize() if task.solutiondef(mosek.soltype.itg): # Output a solution xx = zeros(numvar, Float) task.getsolutionslice(mosek.soltype.itg, mosek.solitem.xx, 0, numvar, xx) print "x =", xx else: print "Integer solution not defined. Probably a problem with 'mosekglb' optimizer." # call the main function try: main () except mosek.Exception, e: print "ERROR: %s" % str(e.errno) if e.msg is not None: print "\t%s" % e.msg sys.exit(1) except: import traceback traceback.print_exc() sys.exit(1)

5.5.1.2. Code comments

Please note that when Task.getsolutionslice is called, the integer solution is requested by using mosek.soltype.itg. No dual solution is defined for integer optimization problems.

5.5.2. Specifying an initial solution

Integer optimization problems are generally hard to solve, but the solution time can often be reduced by providing an initial solution for the solver. Solution values can be set using Task.putsolution (for inputting a whole solution) or Task.putsolutioni (for inputting solution values related to a single variable or constraint).

It is not necessary to specify the whole solution. By setting the mosek.iparam.mio_construct_sol parameter to mosek.onoffkey.on and inputting values for the integer variables only, will force MOSEK to compute the remaining continuous variable values.

If the specified integer solution is infeasible or incomplete, MOSEK will simply ignore it.

5.5.3. Example: Specifying an integer solution

Consider the problem

\begin{math}\nonumber{}\begin{array}{ll}\nonumber{}\mbox{maximize} & 7x_{0}+10x_{1}+x_{2}+5x_{3}\\\nonumber{}\mbox{subject to} & x_{0}+x_{1}+x_{2}+x_{3}\leq{}2.5\\\nonumber{} & x_{0},x_{1},x_{2}\mathrm{integer},\quad{}x_{0},x_{1},x_{2},x_{3}\geq{}0\end{array}\end{math} (5.5.2)

The following example demonstrates how to optimize the problem using a feasible starting solution generated by selecting the integer values as [[MathCmd 65]].

## # Copyright: Copyright (c) 1998-2007 MOSEK ApS, Denmark. All rights reserved. # # File: mioinitsol.cs # # Purpose: Demonstrates how to solve a small mixed # integer linear optimization problem using the MOSEK Python API. ## import sys import pymosek as mosek try: from Numeric import array, Float, zeros, ones except ImportError: from mosekarr import array, Float, zeros, ones # Since the actual value of Infinity is ignores, we define it solely # for symbolic purposes: inf = 0.0 # Define a stream printer to grab output from MOSEK def streamprinter(text): sys.stdout.write(text) sys.stdout.flush() # We might write everything directly as a script, but it looks nicer # to create a function. def main (): # Create a handle to MOSEK mskhandle = mosek.mosek () # Make a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task(0,0) # Attach a printer to the task task.set_Stream (mosek.streamtype.log, streamprinter) bkc = array([ mosek.boundkey.up ]) blc = array([ -inf, ]) buc = array([ 2.5 ]) bkx = array([ mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo ]) blx = array([0.0, 0.0, 0.0, 0.0 ]) bux = array([ inf, inf, inf, inf ]) c = array([ 7.0, 10.0, 1.0, 5.0 ]) asub = array([ 0, 0, 0, 0 ]) acof = array([ 1.0, 1.0, 1.0, 1.0]) ptrb = array([ 0, 1, 2, 3 ]) ptre = array([ 1, 2, 3, 4 ]) numvar = len(bkx) numcon = len(bkc) # Input linear data task.inputdata(numcon,numvar, c,0.0, ptrb, ptre, asub, acof, bkc, blc, buc, bkx, blx, bux) # Input objective sense task.putobjsense(mosek.objsense.maximize) # Define variables to be integers task.putvartypelist(array([ 0, 1, 2 ]), array([ mosek.variabletype.type_int, mosek.variabletype.type_int, mosek.variabletype.type_int])) # Construct an initial feasible solution from the # values of the integer valuse specified task.putintparam(mosek.iparam.mio_construct_sol, mosek.onoffkey.on); # Set status of all variables to unknown task.makesolutionstatusunknown(mosek.soltype.itg); # Assign values 1,1,0 to integer variables task.putsolutioni ( mosek.accmode.var, 0, mosek.soltype.itg, mosek.stakey.supbas, 0.0, 0.0, 0.0, 0.0); task.putsolutioni ( mosek.accmode.var, 1, mosek.soltype.itg, mosek.stakey.supbas, 2.0, 0.0, 0.0, 0.0); task.putsolutioni ( mosek.accmode.var, 2, mosek.soltype.itg, mosek.stakey.supbas, 0.0, 0.0, 0.0, 0.0); # Optimize task.optimize() if task.solutiondef(mosek.soltype.itg): # Output a solution xx = zeros(numvar, Float) task.getsolutionslice(mosek.soltype.itg, mosek.solitem.xx, 0, numvar, xx) print "x =", xx else: print "Integer solution not defined. Probably a problem with 'mosekglb' optimizer." # call the main function try: main () except mosek.Exception, e: print "ERROR: %s" % str(e.errno) if e.msg is not None: print "\t%s" % e.msg sys.exit(1) except: import traceback traceback.print_exc() sys.exit(1)

5.6. Problem modification and reoptimization

Often one might want to solve not just a single optimization problem, but a sequence of problem, each differing only slightly from the previous one. This section demonstrates how to modify and reoptimize an existing problem. The example we study is a simple production planning model.

5.6.1. A production planning problem

A company manufactures three types of products. Suppose the stages of manufacturing can be split into three parts, namely Assembly, Polishing and Packing. In the table below we show the time required for each stage as well as the profit associated with each product.

Product no. Assembly (minutes) Polishing (minutes) Packing (minutes) Profit ($)
0 2 3 2 1.50
1 4 2 3 2.50
2 3 3 2 3.00

With the current resources available, the company has 100,000 minutes of assembly time, 50,000 minutes of polishing time and 60,000 minutes of packing time available per year.

Now the question is how many items of each product the company should produce each year in order to maximize profit?

Denoting the number of items of each type by [[MathCmd 66]] and [[MathCmd 67]], this problem can be formulated as the linear optimization problem:

\begin{math}\nonumber{}\begin{array}{lccccccl}\nonumber{}\mbox{maximize} & 1.5x_{0} & + & 2.5x_{1} & + & 3.0x_{2} &  & \\\nonumber{}\mbox{subject to} & 2x_{0} & + & 4x_{1} & + & 3x_{2} & \leq{} & 100000,\\\nonumber{} & 3x_{0} & + & 2x_{1} & + & 3x_{2} & \leq{} & 50000,\\\nonumber{} & 2x_{0} & + & 3x_{1} & + & 2x_{2} & \leq{} & 60000,\end{array}\end{math} (5.6.1)

and

\begin{math}\nonumber{}x_{0},x_{1},x_{2}\geq{}0.\end{math} (5.6.2)

The following code loads this problem into the optimization task.

# Create a handle to MOSEK mskhandle = mosek.mosek () # Create a MOSEK environment env = mskhandle.Env () # Attach a printer to the environment env.set_Stream (mosek.streamtype.log, streamprinter) # Initialize the environment env.init () # Create a task task = env.Task(0,0) # Attach a printer to the task task.set_Stream (mosek.streamtype.log, streamprinter) # Bound keys for constraints bkc = array ([mosek.boundkey.up, mosek.boundkey.up, mosek.boundkey.up]) # Bound values for constraints blc = array ([-inf, -inf, -inf]) buc = array ([100000.0 , 50000.0, 60000.0]) # Bound keys for variables bkx = array ([mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo]) # Bound values for variables blx = array ([ 0.0, 0.0, 0.0]) bux = array ([+inf, +inf, +inf]) # Objective coefficients csub = array([ 0, 1, 2 ]) cval = array([ 1.5, 2.5, 3.0 ]) # We input the A matrix column-wise # asub contains row indexes asub = array([ 0, 1, 2, 0, 1, 2, 0, 1, 2]) # acof contains coefficients acof = array([ 2.0, 3.0, 2.0, 4.0, 2.0, 3.0, 3.0, 3.0, 2.0 ]) # aptrb and aptre contains the offsets into asub and acof where # columns start and end respectively aptrb = array([ 0, 3, 6 ]) aptre = array([ 3, 6, 9 ]) numvar = len(bkx) numcon = len(bkc) # Append the constraints task.append(mosek.accmode.con,numcon) # Append the variables. task.append(mosek.accmode.var,numvar) # Input objective task.putcfix(0.0) task.putclist(csub,cval) # Put constraint bounds task.putboundslice(mosek.accmode.con, 0, numcon, bkc, blc, buc) # Put variable bounds task.putboundslice(mosek.accmode.var, 0, numvar, bkx, blx, bux) # Input A non-zeros by columns for j in range(numvar): ptrb,ptre = aptrb[j],aptre[j] task.putavec(mosek.accmode.var,j, asub[ptrb:ptre], acof[ptrb:ptre]) # Input the objective sense (minimize/maximize) task.putobjsense(mosek.objsense.maximize) # Optimize the task task.optimize() # Output a solution xx = zeros(numvar, Float) task.getsolutionslice(mosek.soltype.bas, mosek.solitem.xx, 0,numvar, xx) print "xx =", xx

5.6.2. Changing the A matrix

Suppose we want to change the time required for assembly of product 0 to 3 minutes. This corresponds to setting [[MathCmd 70]], which is done by calling the function Task.putaij as shown below.

task.putaij(0, 0, 3.0)

The problem now has the form:

\begin{math}\nonumber{}\begin{array}{lccccccl}\nonumber{}\mbox{maximize} & 1.5x_{0} & + & 2.5x_{1} & + & 3.0x_{2} &  & \\\nonumber{}\mbox{subject to} & 3x_{0} & + & 4x_{1} & + & 3x_{2} & \leq{} & 100000,\\\nonumber{} & 3x_{0} & + & 2x_{1} & + & 3x_{2} & \leq{} & 50000,\\\nonumber{} & 2x_{0} & + & 3x_{1} & + & 2x_{2} & \leq{} & 60000,\end{array}\end{math} (5.6.3)

and

\begin{math}\nonumber{}x_{0},x_{1},x_{2}\geq{}0.\end{math} (5.6.4)

After changing the A matrix we can find the new optimal solution by calling Task.optimize again.

5.6.3. Appending variables

We now want to add a new product with the following data:

Product no. Assembly (minutes) Polishing (minutes) Packing (minutes) Profit ($)
3 4 0 1 1.00

This corresponds to creating a new variable [[MathCmd 73]], appending a new column to the A matrix and setting a new value in the objective. We do this in the following code.

# Append a new varaible x_3 to the problem */ task.append(mosek.accmode.var,1) # Set bounds on new varaible task.putbound(mosek.accmode.var, task.getnumvar()-1, mosek.boundkey.lo, 0, +inf) # Change objective task.putcj(task.getnumvar()-1,1.0) # Put new values in the A matrix acolsub = array([0, 2]) acolval = array([4.0, 1.0]) task.putavec(mosek.accmode.var, task.getnumvar()-1, # column index acolsub, acolval)

After this operation the problem looks this way:

\begin{math}\nonumber{}\begin{array}{lccccccccl}\nonumber{}\mbox{maximize} & 1.5x_{0} & + & 2.5x_{1} & + & 3.0x_{2} & + & 1.0x_{3} &  & \\\nonumber{}\mbox{subject to} & 3x_{0} & + & 4x_{1} & + & 3x_{2} & + & 4x_{3} & \leq{} & 100000,\\\nonumber{} & 3x_{0} & + & 2x_{1} & + & 3x_{2} &  &  & \leq{} & 50000,\\\nonumber{} & 2x_{0} & + & 3x_{1} & + & 2x_{2} & + & 1x_{3} & \leq{} & 60000,\end{array}\end{math} (5.6.5)

and

\begin{math}\nonumber{}x_{0},x_{1},x_{2},x_{3}\geq{}0.\end{math} (5.6.6)

5.6.4. Reoptimization

When Task.optimize is called MOSEK will store the optimal solution internally. After a task has been modified and Task.optimize is called again the solution will automatically be used to reduce solution time of the new problem, if possible.

In this case an optimal solution to problem (5.6.3) was found and then added a column was added to get (5.6.5). The simplex optimizer is well suited for exploiting an existing primal or dual feasible solution. Hence, the subsequent code instructs MOSEK to choose the simplex optimizer freely when optimizing.

# Change optimizer to simplex free and reoptimize task.putintparam(mosek.iparam.optimizer,mosek.optimizertype.free_simplex) task.optimize()

5.6.5. Appending constraints

Now suppose we want to add a new stage to the production called “Quality control” for which 30000 minutes are available. The time requirement for this stage is shown below:

Product no. Quality control (minutes)
0 1
1 2
2 1
3 1

This corresponds to adding the constraint

\begin{math}\nonumber{}x_{0}+2x_{1}+x_{2}+x_{3}\leq{}30000\end{math} (5.6.7)

to the problem which is done in the following code:

# Append a new constraint task.append(mosek.accmode.con,1) # Set bounds on new constraint task.putbound( mosek.accmode.con, task.getnumcon()-1, # row index mosek.boundkey.up, -inf, 30000) # Put new values in the A matrix arowsub = array([0, 1, 2, 3 ]) arowval = array([1.0, 2.0, 1.0, 1.0]) task.putavec(mosek.accmode.con, task.getnumcon()-1, # row index arowsub, arowval)

5.7. Efficiency considerations

Although MOSEK is implemented to handle memory efficiently, the user may have valuable knowledge about a problem, which could be used to improve the performance of MOSEK. This section discusses some tricks and general advice that hopefully make MOSEK process your problem faster.

Avoid memory fragmentation:

MOSEK stores the optimization problem in internal data structures in the memory. Initially MOSEK will allocate structures of a certain size, and as more items are added to the problem the structures are reallocated. For large problems the same structures may be reallocated many times causing memory fragmentation. One way to avoid this is to give MOSEK an estimated size of your problem using the functions:

None of these functions change the problem, they only give hints to the eventual dimension of the problem. If the problem ends up growing larger than this, the estimates are automatically increased.

Tune the reallocation process:

It is possible to obtain information about how often MOSEK reallocates storage for the A matrix by inspecting mosek.iinfitem.sto_num_a_realloc. A large value indicates that maxnumanz has been reestimated many times and that the initial estimate should be increased.

Do not mix put- and get- functions:

For instance, the functions Task.putavec and Task.getavec. MOSEK will queue put- commands internally until a get- function is called. If every put- function call is followed by a get- function call, the queue will have to be flushed often, decreasing efficiency.

In general get- commands should not be called often during problem setup.

Use the LIFO principle when removing constraints and variables:

MOSEK can more efficiently remove constraints and variables with a high index than a small index.

An alternative to removing a constraint or a variable is to fix it at 0, and set all relevant coefficients to 0. Generally this will not have any impact on the optimization speed.

Add more constraints and variables than you need (now):

The cost of adding one constraint or one variable is about the same as adding many of them. Therefore, it may be worthwhile to add many variables instead of one. Initially fix the unused variable at zero, and then later unfix them as needed. Similarly, you can add multiple free constraints and then use them as needed.

Use one environment (env) only:

If possible share the environment (env) between several tasks. For most applications you need to create only a single env.

Do not remove basic variables:

When doing reoptimizations, instead of removing a basic variable it may be more efficient to fix the variable at zero and then remove it when the problem is reoptimized and it has left the basis. This makes it easier for MOSEK to restart the simplex optimizer.

5.7.1. API overhead

The Python interface is a thin wrapper around a native MOSEK library. The layer between the Python application and the native MOSEK library is made as thin as possible to minimize the overhead from function calls.

The methods in mosek.Env and mosek.Task are all written in C and resides in the module pymosek. Each method converts the call parameter data structures (i.e. creates a complete copy of the data), calls a MOSEK function and converts the returned values back into Python structures.

All data are copied at least once. For larger problems this may mean, that fetching or inputting large chunks of data is less expensive than fetching/inputting the same data as single values.

5.8. Conventions employed in the API

5.8.1. Naming conventions for arguments

In the definition of the MOSEK Python API a consistent naming convention has been used. This implies that whenever for example numcon is an argument in a function definition it indicates the number of constraints.

In Table 5.2 the variable names used to specify the problem parameters are listed.

Python name Python type Dimension Related problem
  parameter
numcon int m
numvar int n
numcone int t
numqonz int [[MathCmd 77]]
qosubi int[] numqonz [[MathCmd 77]]
qosubj int[] numqonz [[MathCmd 77]]
c float[] numvar [[MathCmd 80]]
cfix float [[MathCmd 5]]
numqcnz int [[MathCmd 82]]
qcsubk int[] qcnz [[MathCmd 82]]
qcsubi int[] qcnz [[MathCmd 82]]
qcsubj int[] qcnz [[MathCmd 82]]
aptrb int[] numvar [[MathCmd 86]]
aptre int[] numvar [[MathCmd 86]]
asub int[] aptre[numvar-1] [[MathCmd 86]]
aval float[] aptre[numvar-1] [[MathCmd 86]]
blc float[] numcon [[MathCmd 90]]
buc float[] numcon [[MathCmd 91]]
blx float[] numvar [[MathCmd 92]]
bux float[] numvar [[MathCmd 93]]
Table 5.2: Naming convention used in MOSEK. Here TTT[] means a Numeric.array of type TTT.

The relation between the variable names and the problem parameters is as follows:

  • The quadratic terms in the objective:

    \begin{math}\nonumber{}q_{{\mathtt{qosubi[t]},\mathtt{qosubj[t]}}}^{o}=\mathtt{qoval[t]},~t=0,\ldots ,\mathtt{numqonz}-1.\end{math} (5.8.1)
  • The linear terms in the objective:

    \begin{math}\nonumber{}c_{j}=\mathtt{c[j]},~j=0,\ldots ,\mathtt{numvar}-1\end{math} (5.8.2)
  • The fixed term in the objective:

    \begin{math}\nonumber{}c^{f}=\mathtt{cfix}.\end{math} (5.8.3)
  • The quadratic terms in the constraints:

    \begin{math}\nonumber{}q_{{\mathtt{qcsubi[t]},\mathtt{qcsubj[t]}}}^{\mathtt{qcsubk[t]}}=\mathtt{qcval[t]},~t=0,\ldots ,\mathtt{numqcnz}-1.\end{math} (5.8.4)
  • The linear terms in the constraints:

    \begin{math}\nonumber{}\begin{array}{rl}\nonumber{}a_{{\mathtt{asub[t],j}}}=\mathtt{aval[t]}, & t=\mathtt{ptrb[j]},\ldots ,\mathtt{ptre[j]}-1,\\\nonumber{} & j=0,\ldots ,\mathtt{numvar}-1.\end{array}\end{math} (5.8.5)
  • The bounds on the constraints are specified using the variables bkc, blc, and buc. The components of the integer array bkc specify the bound type according to Table 5.3.

    Symbolic constant Lower bound Upper bound
    mosek.boundkey.fx finite identical to the lower bound
    mosek.boundkey.fr minus infinity plus infinity
    mosek.boundkey.lo finite plus infinity
    mosek.boundkey.ra finite finite
    mosek.boundkey.up minus infinity finite
    Table 5.3: Interpretation of the bound keys.

    For instance bkc[2]=mosek.boundkey.lo means that [[MathCmd 18]] and [[MathCmd 19]]. Finally, the numerical values of the bounds are given by

    \begin{math}\nonumber{}l_{k}^{c}=\mathtt{blc[k]},~k=0,\ldots ,\mathtt{numcon}-1\end{math} (5.8.6)

    and

    \begin{math}\nonumber{}u_{k}^{c}=\mathtt{buc[k]},~k=0,\ldots ,\mathtt{numcon}-1.\end{math} (5.8.7)
  • The bounds on the variables are specified using the variables bkx, blx, and bux. The components in the integer array bkx specify the bound type according to Table 5.3. The numerical values for the lower bounds on the variables are given by

    \begin{math}\nonumber{}l_{j}^{x}=\mathtt{blx[j]},~j=0,\ldots ,\mathtt{numvar}-1.\end{math} (5.8.8)

    The numerical values for the upper bounds on the variables are given by

    \begin{math}\nonumber{}u_{j}^{x}=\mathtt{bux[j]},~j=0,\ldots ,\mathtt{numvar}-1.\end{math} (5.8.9)

5.8.1.1. Bounds

A bound on a variable or on a constraint in MOSEK consists of a bound key, as defined in Table 5.3, a lower bound value and an upper bound value. Even if a variable or constraint is bounded only from below, e.g. x0, both bounds are inputted or extracted; the value inputted as upper bound for (x0) is ignored.

5.8.2. Vector formats

Three different vector formats are used in the MOSEK API:

Full vector:

This is simply an array where the first element corresponds to the first item, the second element to the second item etc. For example to get the linear coefficients of the objective in task, one would write

c = zeros(numvar,Float) task.getc(c)

where numvar is the number of variables in the problem.

Vector slice:

A vector slice is a range of values. For example, to get the bounds associated constraint 3 through 10 (both inclusive) one would write

upper_bound = zeros(8,Float) lower_bound = zeros(8,Float) bound_key = array([None] * 8) task.getboundslice(accmode.con, 2,10, bound_key,lower_bound,upper_bound)

Please note that items in MOSEK are numbered from 0, so that the index of the first item is 0, and the index of the n'th item is n-1.

Sparse vector:

A sparse vector is given as an array of indexes and an array of values. For example, to input a set of bounds associated with constraints number 1, 6, 3, and 9, one might write

bound_index = array([ 1, 6, 3, 9]) bound_key = array([boundkey.fr,boundkey.lo,boundkey.up,boundkey.fx]) upper_bound = array([ 0.0, -10.0, 0.0, 5.0]) lower_bound = array([ 0.0, 0.0, 6.0, 5.0]) task.putboundlist(accmode.con, bound_index, bound_key,lower_bound,upper_bound)

Note that the list of indexes need not be ordered.

5.8.3. Matrix formats

The coefficient matrices in a problem are inputted and extracted in a sparse format, either as complete or a partial matrices. Basically there are two different formats for this.

5.8.3.1. Unordered triplets

In unordered triplet format each entry is defined as a row index, a column index and a coefficient. For example, to input the A matrix coefficients for [[MathCmd 105]], [[MathCmd 106]], and [[MathCmd 107]], one would write as follows:

subi = array([ 1, 3, 5 ]) subj = array([ 2, 3, 4 ]) cof = array([ 1.1, 4.3, 0.2 ]) task.putaijlist(subi,subj,cof)

Please note that in some cases (like Task.putaijlist) only the specified indexes remain modified — all other are unchanged. In other cases (such as Task.putqconk) the triplet format is used to modify all entries — entries that are not specified are set to 0.

5.8.3.2. Row or column ordered sparse matrix

In a sparse matrix format only the non-zero entries of the matrix are stored. MOSEK uses a sparse matrix format ordered either by rows or columns. In the column-wise format the position of the non-zeros are given as a list of row indexes. In the row-wise format the position of the non-zeros are given as a list of column indexes. Values of the non-zero entries are given in column or row order.

A sparse matrix in column ordered format consists of:

asub:

List of row indexes.

aval:

List of non-zero entries of A ordered by columns.

ptrb:

Where ptrb[j] is the position of the first value/index in aval / asub for column j.

ptre:

Where ptre[j] is the position of the last value/index plus one in aval / asub for column j.

The values of a matrix A with numcol columns are assigned so that for

\begin{displaymath}\nonumber{}j=0,\ldots ,\mathtt{numcol}-1.\end{displaymath}

We define

\begin{math}\nonumber{}\begin{array}{rcl}\nonumber{}a_{{\mathtt{asub}[k],j}}=\mathtt{aval}[k],\quad{}k=\mathtt{ptrb}[j],\ldots ,\mathtt{ptre}[j]-1.\end{array}\end{math} (5.8.10)

Figure 5.1: The matrix A (5.8.11) represented in column ordered sparse matrix format.

As an example consider the matrix

\begin{math}\nonumber{}A=\left[\begin{array}{ccccc}\nonumber{}1.1 &  & 1.3 & 1.4 & \\\nonumber{} & 2.2 &  &  & 2.5\\\nonumber{}3.1 &  &  & 3.4 & \\\nonumber{} &  & 4.4 &  &\end{array}\right].\end{math} (5.8.11)

which can be represented in the column ordered sparse matrix format as

\begin{displaymath}\nonumber{}\begin{array}{lcl}\nonumber{}\mathtt{ptrb} & = & [0,2,3,5,7],\\\nonumber{}\mathtt{ptre} & = & [2,3,5,7,8],\\\nonumber{}\mathtt{asub} & = & [0,2,1,0,3,0,2,1],\\\nonumber{}\mathtt{aval} & = & [1.1,3.1,2.2,1.3,4.4,1.4,3.4,2.5].\end{array}\end{displaymath}

Fig. 5.1 illustrates how the matrix A (5.8.11) is represented in column ordered sparse matrix format.

5.8.3.3. Row ordered sparse matrix

The matrix A (5.8.11) can also be represented in the row ordered sparse matrix format as:

\begin{displaymath}\nonumber{}\begin{array}{lcl}\nonumber{}\mathtt{ptrb} & = & [0,3,5,7],\\\nonumber{}\mathtt{ptre} & = & [3,5,7,8],\\\nonumber{}\mathtt{asub} & = & [0,2,3,1,4,0,3,2],\\\nonumber{}\mathtt{aval} & = & [1.1,1.3,1.4,2.2,2.5,3.1,3.4,4.4].\end{array}\end{displaymath}

5.8.4. Array objects

The MOSEK Python API provides a simple array object in the module mosekarr. This includes a one-dimensional dense array which can be of type Float, Int or Object, and a few operators and functions to create and modify array objects.

Arrays can be constructed in several ways:

import mosekarr # Create an array of integers a0 = mosekarr.array([1,2,3],mosekarr.Int) # Create an array of floats a1 = mosekarr.array([1,2,3],mosekarr.Float) # Create an integer array of ones a2 = mosekarr.ones(10) # Create an float array of ones a3 = mosekarr.ones(10,mosekarr.Float) # Create a range of integers 5,6,...,9 a4 = mosekarr.range(5,10) # Create and array of objects a5 = mosekarr.array(['a string', 'b string', 10, 2.2])

A limited set of operations on arrays are available - these should work more or less like the equivalent Numeric operations:

a = ones(10,mosekarr.Float) b = 1.0 * range(10) # element-wise multiplication, addition and subtraction c0 = a * b c1 = a + b c2 = a - b # multiplly each element by 2.1 c4 = a * 2.1 # add 2 to each element c5 = a + 2

Finally, if the Numeric package is available, arrays can be converted between Numeric and mosekarr types:

A = Numeric.array(mosekarr.ones(10)) B = mosekarr.array(Numeric.ones(10))

If more advanced array operations is needed, it is necessary to install the Python Numeric package.

5.8.5. Typical problems using the Python API

Since all all type-information in Python is implicit, type-checking is performed only when required, and in certain cases it is necessary to explicitly write type information.

The MOSEK API currently only supports its own array object and Python Numeric arrays (not from the standard array module), and it does not implicitly convert arrays. This means that if MOSEK in some function call expects an array of floats, it will not accept, say, an array of ints. The MOSEK Python default array object is found in the module mosekarr which contains a minimal subset of functionality similar to Numeric.

Typically type errors occur in two situations:

  • The array-type does not match the expected, and an ValueError exception is raised with a message

    In function XX: Argument <YY> should be of type array
    

    Note in particular that the python module MA (Matlab compatibility functions) in some cases produce Python arrays, not Python Numeric arrays. The type of a variable xx can always be displayed by the line

    print type(xx)
    
  • The array had the correct type, but the elements in the array did not have the expected type. This will result in a ValueError exception with a message

    In function XX: Array argument <YY> has wrong type
    

    This may be due to implicit typing; for example an array

    Numeric.array([1,2,3])
    

    will become an array of integers while

    Numeric.array([1.0,2.0,3.0])
    

    is an array of floats. To avoid this kind of errors, it is recommended that the array type is explicitly provided, for example as

    Numeric.array([1.0,2.0,3.0], Numeric.Float)
    

    The type-code of an array xx can be displayed as

    print xx.typecode
    

    The specific meanings of type-codes are listed in the Python Numeric documentation.

Mon Sep 14 15:53:06 2009