optimization_in_scilab.pdf

Txt a string providing information on what the user

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txt : a string providing information on what the user should do next. In this mode, lmitool generates a file in the current directory. The name of this file is obtained by adding “.sci” to the end of probname . This file is the skeleton of a solver function and the corresponding evaluation function. Example Suppose we want to use lmitool to solve the problem presented in Section 7.2.2 . Invoking -->txt=lmitool(’sf_sat’,’Q,Y’,’A,B,umax’) yields the output --> txt = ! To solve your problem, you need to ! ! ! !1- edit file /usr/home/DrScilab/sf_sat.sci ! ! ! !2- load (and compile) your functions: ! ! ! ! getf(’/usr/home/DrScilab/sf_sat.sci’,’c’) ! 49
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! ! !3- Define A,B,umax and call sf_sat function: ! ! ! ! [Q,Y]=sf_sat(A,B,umax) ! ! ! !To check the result, use [LME,LMI,OBJ]=sf_sat_eval(list(Q,Y)) ! and results in the creation of the file ’/usr/home/curdir/sf sat.sci’ with the following content: function [Q,Y]=sf_sat(A,B,umax) // Generated by lmitool on Tue Feb 07 10:30:35 MET 1995 Mbound = 1e3; abstol = 1e-10; nu = 10; maxiters = 100; reltol = 1e-10; options=[Mbound,abstol,nu,maxiters,reltol]; ///////////DEFINE INITIAL GUESS BELOW Q_init=... Y_init=... /////////// XLIST0=list(Q_init,Y_init) XLIST=lmisolver(XLIST0,sf_sat_eval,options) [Q,Y]=XLIST(:) /////////////////EVALUATION FUNCTION//////////////////////////// function [LME,LMI,OBJ]=sf_sat_eval(XLIST) [Q,Y]=XLIST(:) /////////////////DEFINE LME, LMI and OBJ BELOW LME=... LMI=... OBJ=... It is easy to see how a small amount of editing can do the rest! 7.3.2 Interactive mode lmitool can be invoked with zero or one input argument as follows: 50
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Syntax txt=lmitool() txt=lmitool(file) where file : is a string giving the name of an existing “.sci” file generated by lmitool . In this mode, lmitool is fully interactive. Using a succession of dialogue boxes, user can com- pletely define his problem. This mode is very easy to use and its operation completely self explanatory. Invoking lmitool with one argument allows the user to start off with an existing file. This mode is useful for modifying existing files or when the new problem is not too much different from a problem already treated by lmitool . Example Consider the following estimation problem y = Hx + V w where x is unknown to be estimated, y is known, w is a unit-variance zero-mean Gaussian vector, and H Co { H (1) , ..., H ( N ) } , V Co { V (1) , ..., V ( N ) } where Co denotes the convex hull and H ( i ) and V ( i ), i = 1 , ..., N, are given matrices. The objective is to find L such that the estimate ˆ x = Ly is unbiased and the worst case estimation error variance E( k x - ˆ x k 2 ) is minimized. It can be shown that this problem can be formulated as a Σ problem as follows: minimize γ subject to I - LH ( i ) = 0 , i = 1 , ..., N, X ( i ) - X ( i ) T = 0 , i = 1 , ..., N, and I ( L ( i ) V ( i )) T L ( i ) V ( i ) X ( i ) 0 , i = 1 , ..., N, γ - Trace( X ( i )) 0 , i = 1 , ..., N. To use lmitool for this problem, we invoke it as follows: --> lmitool() This results is an interactive session which is partly illustrated in following figures. 51
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7.4 How lmisolver works The function lmisolver works essentially in four steps: 1. Initial set-up. The sizes and structure of the initial guess are used to set up the problem, and in particular the size of the unknown vector.
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