optimization_in_scilab.pdf

2 elimination of equality constraints making repeated

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2. Elimination of equality constraints. Making repeated calls to the evaluation function, lmisolver generates a canonical representation of the form minimize ˜ c T z subject to ˜ F 0 + z 1 ˜ F 1 + · · · + z ˜ m ˜ F ˜ m 0 , Az + b = 0 , where z contains the coefficients of all matrix variables. This step uses extensively sparse matrices to speed up the computation and reduce memory requirement. 3. Elimination of variables. Then, lmisolver eliminates the redundant variables. The equality constraints are eliminated by computing the null space N of A and a solution z 0 (if any) of Ax + b = 0. At this stage, all solutions of the equality constraints are parametrized by z = Nx + z 0 , where x is a vector containing the independent variables. The computation of N, z 0 is done using sparse LU functions of Scilab. Once the equality constraints are eliminated, the problem is reformulated as minimize c T x subject to F 0 + x 1 F 1 + · · · + x m F m 0 , where c is a vector, and F 0 , . . . , F m are symmetric matrices, and x contains the indepen- dent elements in the matrix variables X 1 , . . . , X M . (If the F i ’s are dependent, a column compression is performed.) Figure 7.1: This window must be edited to define problem name and the name of variables used. 52
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Figure 7.2: For the example at hand the result of the editing should look something like this. 4. Optimization. Finally, lmisolver makes a call to the function semidef (an interface to SP [ 23 ]). This phase is itself divided into a feasibility phase and a minimization phase (only if the linear objective function is not empty). The feasibility phase is avoided if the initial guess is found to be feasible. The function semidef is called with the optimization parameters abstol , nu , maxiters , reltol . The parameter M is set above the value Mbnd*max(sum(abs([F0 ... Fm]))) For details about the optimization phase, and the meaning of the above optimization pa- rameters see manual page for semidef . 7.5 Other versions LMITOOL is also available on Matlab. The Matlab version can be obtained by anonymous ftp from ftp.ensta.fr under /pub/elghaoui/lmitool . 53
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Figure 7.3: This is the skeleton of the solver function and the evaluation function generated by LMITOOL using the names defined previously. 54
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Figure 7.4: After editing, we obtain. Figure 7.5: A file is proposed in which the solver and evaluation functions are to be saved. You can modify it if you want. 55
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Chapter 8 Optimization data files This section presents the optimization data files which can be used to configure a specific opti- mization problem in Scilab. The following is a (non-exhaustive) list of ASCII file formats often used in optimization softwares : SIF : Standard Input Format [ 1 , 30 ], GAMS : General Algebraic Modeling System [ 40 , 16 ] AMPL : A Mathematical Programming Language [ 10 , 39 ] MPS : Mathematical Programming System [ 27 , 41 ] but other file formats appeared in recent years, such as the XML-based file format OSiL [ 35 , 8 , 36 ].
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