an Open Source MATLAB- Based Optimization Tool By Amila - - PowerPoint PPT Presentation

CVX an Open Source MATLAB- Based Optimization Tool

By Amila Tharaperiya Gamage Winter 2012

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About CVX

 CVX is an open source MATLAB-based

modeling tool.

 The optimization problem has to be a

convex optimization problem.

 E.g.,

• linear programs (LPs)
• quadratic programs (QPs)
• entropy maximization, etc

 CVX is not for large scale problems.

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CVX: http://cvxr.com/cvx/

About CVX

 Core solvers used in CVX:

• SeDuMi (http://sedumi.ie.lehigh.edu/ )
• SDPT3

(http://www.math.nus.edu.sg/~mattohkc/sdpt3.html)

 Both are open-source interior-point solvers

based on MATLAB.

 CVX converts the problem into a format

accepted by those solvers and call them to solve the problem.

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Disciplined Convex Programming (DCP)

 DCP is a methodology for constructing

convex optimization problems in a suitable format for CVX.

 DCP imposes a set of rules.  Problems has to be written such that

those rules are satisfied. Otherwise, problem will be rejected, even when the problem is convex.

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Functions

 Convexity or concavity has to be followed

from some DCP rules.

• Write the functions in terms of CVX recognized

functions.

 Valid compositions of functions must be

used (refer to CVX user guide for details)

 Product of variables is not convex.

However, geometric mean of variables is a concave function.

• geo_mean([x1 , x2 , x3])= (x1x2x3)1/3

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User guide: http://web.cvxr.com/cvx/cvx_usrguide.pdf

Constraints

 Equality == :Both left- and right-hand

sides should be affine functions of the

• ptimization variables.

 Less-than <= :Expression in left-hand

side should be convex, and the right-hand expression should be concave.

 Constraints are imposed elementwise for

arrays and matrices.

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affine: f(ax + (1 − a)y)= af(x) + (1 − a )f(y)

What are the problems CVX could not handle well?

 Log(x), exp(x), log(exp x1+· · ·+exp xn)  CVX uses successive approximation

method for solving these problems, and the results may not be accurate.

 Solution: try to convert the problem to

some other form if possible

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Some Useful Conversions

 Use Max sqrt(x1x2) instead of Max log(x1)+log(x2)

 Max log(1+xy/(x+y))

subject to x,y>0 :

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Max 1+t Subject to sqrt[(x-t)(y-t)] >= t x,y,t>0 : Max log(1+t) Subject to sqrt[(x-t)(y-t)] >= t x,y,t>0 :

xy/(x+y) >= t

CVX commands

 cvx_optval: provide the optimum value of

the objective function

 cvx_slvtol: the tolerance level achieved by

the solver

 cvx_slvitr: the number of iterations taken

by the solver

 cvx_precision: modify the tolerances of

the solver

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System Model for Demo

Macrocell network Macrocell user

FAP Relay-1 Relay-2 Destination-1 Destination-2

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Thank you

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