PENLIB/SDP User s Guide

Transcription

1 PENLIB/SDP User s Guide Michal Kočvara Michael Stingl In this guide we give a description of parameters of function sdp, solving linear semidefinite programming problems with linear constraints. This function is part of the library PENLIB. We solve the dual linear SDP problem with linear constraints: n f i x i min x R n s.t. i= i + n b k i x k c i, k= i =,..., m l n k i x k i =,..., m. k= The matrix constraints can be written as one constraint with block diagonal matrices as follows: m + + m m x x n n m n x n Here we use the abbreviations n := vars, m l := constr, and m := mconstr. The input parameters are explained below. We assume that the linear constraint vectors b i can be sparse, so we give them in standard sparse format. Similarly, we assume that the matrix constraints data can be sparse. Here we distinguish two cases: Some many of the matrices i k for the k th constraint can be empty; so we only give those matrices for each matrix constraint that are nonempty. Further, each of the nonempty matrices i k can still be sparse; hence we give the matrices i k in sparse format value, column index, row index. Further, as all the matrices are symmetric, we only give the upper triangle. The function sdp is declared as int sdpint vars, int constr, int mconstr, int* msizes, double *fx, double* x, double* uoutput, double* fobj, double* ci, int* bi_dim, int* bi_idx, double* bi_val, int* ai_dim, int* ai_idx, int* ai_nzs, double* ai_val, int* ai_col, int* ai_row, int* ioptions, double* foptions, int* iresults, double* fresults, int* info;

2 vars number of variables integer number constr number of linear constraints integer number 3 mconstr number of matrix constraints diagonal blocks in each k integer number 4 msizes sizes of the diagonal blocks k, k,..., mconstr k integer array length: mconstr 5 fx on exit: objective function value double array length: 6 x on entry: initial guess for the solution on exit: solution vector x Not referenced, if x = on entry double array length: vars 7 uoutput on exit: linear multipliers u i i =,..., constr followed by upper triangular parts of matrix multipliers U j j =,..., mconstr in rowise storage format Not referenced, if uoutput = on entry double array length: constr + msizes*msizes+/ + msizes*msizes+/ msizesmconstr*msizesmconstr+/ 8 fobj objective vector f in full format double array length: vars 9 ci right-hand side vector of the linear constraint c in full format double array length: constr bi dim number of nonzeros in vector b i for each linear constraint integer array length: constr bi idx indices of nonzeros in each vector b i integer array length: bi dim+bi dim+... +bi dimconstr bi val nonzero values in each vector b i corresponding to indices in bi idx double array length: bi dim+bi dim+... +bi dimconstr 3 ai dim number of nonzero blocks i, i,..., i vars for each matrix constraint i =,,..., mconstr integer array length: mconstr 4 ai idx indices of nonzero blocks for each matrix constraint integer array length: ai dim+ai dim+... +ai dimmconstr 5 ai nzs number of nonzero values in each nonzero block i ai idx, i ai idx,..., i ai idxai dimi for each matrix constraint i =,,..., mconstr integer array length: ai dim+ai dim+... +ai dimmconstr 6 ai val nonzero values in the upper triangle of each nonzero block i ai idx, i ai idx,..., i ai idxai dimi for each matrix constraint i =,,..., mconstr double array length: ai nzs+ai nzs+... +ai nzslengthai nzs 7 ai col column indices of nonzero values in the upper triangle of each nonzero block i ai idx, i ai idx,..., i ai idxai dimi for each matrix constraint i =,,..., mconstr integer array length: ai nzs+ai nzs+... +ai nzslengthai nzs 8 ai row row indices of nonzero values in the upper triangle of each nonzero block i ai idx, i ai idx,..., i ai idxai dimi for each matrix constraint i =,,..., mconstr integer array length: ai nzs+ai nzs+... +ai nzslengthai nzs

3 9 ioptions integer valued options see below integer array length: 8 foptions real valued options see below double array length: 7 iresults on exit: integer valued output information see below Not referenced, if iresults = on entry integer array length: 4 fresults on exit: real valued output information see below Not referenced, if fresults = on entry double array length: 3 3 info on exit: error flag see below integer array length: IOPTIONS name/value meaning default ioption DEF use default values for all options use user defined values ioption PBM MX ITER maximum number of iterations of the overall 5 algorithm ioption UM MX ITER maximum number of iterations for the unconstrained minimization ioption3 OUTPUT output level no output summary output brief output 3 full output ioption4 DENSE check density of the Hessian automatic check. For very large problems with dense Hessian, this may lead to memory difficulties. dense Hessian assumed ioption5 LS linesearch in unconstrained minimization do not use linesearch use linesearch ioption6 XOUT write solution vector x in the output file no yes ioption7 UOUT write computed multipliers in the output file no yes FOPTIONS name/value meaning default foption U scaling factor for linear constraints; must be positive. foption MU restriction for multiplier update for linear constraints.7 foption MU restriction for multiplier update for matrix constraints. foption3 PBM EPS stopping criterium for the overall algorithm.e-7 foption4 P EPS lower bound for the penalty parameters.e-6 foption5 UMIN lower bound for the multipliers.e-4 foption6 LPH stopping criterium for unconstrained minimization.e- 3

4 IRESULTS iresults iresults iresults iresults3 meaning number of outer iterations number of Newton steps number of linesearch steps elapsed time in seconds FRESULTS fresults fresults fresults meaning relative precision at x opt feasibility of linear inequalities at x opt feasibility of matrix inequalities at x opt INFO info info info info3 info4 info5 info6 info7 meaning no errors occured no progress in objective value, problem probably infeasible. Cholesky factorization of Hessian failed. The result still may be useful. Maximum iteration limit exceeded. The result still may be useful. Linesearch failed. The result still may be useful. Wrong input parameters ioptions, foptions. Memory error. Unknown error, please contact PENOPT GbR contact@penopt.com. 4

5 Example. Let f =,, 3 T. ssume that we have no linear constraints. ssume further that we have two matrix inequality constraints, first of size 3x3, second of size x. The first constraint contains full matrices, the second one diagonal matrices: + x + x x 3 The blocks k have then 6 nonzero elements, block k only two nonzero elements recall that we only give elements of the upper triangular matrix. In this case vars = 3 constr = mconstr = msizes = 3, x =.,.,. for example fobj =.,.,3. ci =. bi dim = bi idx = bi val =. ai dim = 4,4 ai idx =,,,3,,,,3 ai nzs = 6,6,6,6,,,, ai val =,..., 6,,..., 6,......,,,,,,, 3, 3 ai col =,,,,,,,,,,,,......,,,,,,,, ai row =,,,,,,,,,,,,......,,,,,,,, 5

6 Example. Let again f =,, 3 T. We have two linear constraints with b =,, T, b = 5, 6, T, c = 3, 3 T ssume further that we have two matrix inequality constraints, first of size 3x3, second of size x. The first constraint contains sparse matrices, the second one diagonal matrices. Some of the matrices are empty, as shown below: x + x x 3 In this case vars = 3 constr = mconstr = msizes = 3, x =.,.,. for example fobj =.,.,3. ci = 3., -3. bi dim =, bi idx =,, bi val =., 5., 6. ai dim =,3 ai idx =,3,,, ai nzs = 4,4,,, ai val =.,-.,.,.,.,-.,.,.,.,.,-., 3.,-3. ai col =,,,,,,,,,,,, ai row =,,,,,,,,,,,, 6