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 1

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. CVX: http://cvxr.com/cvx/ 2

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. 3

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. 4

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([ x 1 , x 2 , x 3 ])= ( x 1 x 2 x 3 ) 1/3 User guide: http://web.cvxr.com/cvx/cvx_usrguide.pdf 5

Constraints  Equality == :Both left- and right-hand sides should be affine functions of the optimization 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. affine: f(ax + (1 − a)y)= af (x) + (1 − a )f(y) 6

What are the problems CVX could not handle well?  Log(x), exp(x), log(exp x 1 +· · ·+exp x n )  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 7

Some Useful Conversions  Use Max sqrt( x 1 x 2 ) instead of Max log( x 1 )+log( x 2 )  Max log(1+xy/(x+y)) subject to x,y>0 : xy/(x+y) >= t Max log(1+t) Max 1+t Subject to Subject to sqrt[(x-t)(y-t)] >= t sqrt[(x-t)(y-t)] >= t x,y,t>0 x,y,t>0 : : 8

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 9

System Model for Demo Relay-1 Destination-1 FAP Macrocell network Destination-2 Relay-2 Macrocell user 10

Thank you 11