panagiotisrep/sparse_sdp

Collection of sparse linear SDP problems arising in structural optimization.

SPARSE SDP

This GitHub repository is a "modern" presentation of the original "SPARSE SDP" library by Michal Kočvara hosted at http://web.mat.bham.ac.uk/kocvara/pennon/problems.html.

On the famous page of Hans D. Mittelmann, this library is called "KOCVARA" http://plato.asu.edu/ftp/kocvara/.

This is a collection of sparse linear SDP problems arising in structural optimization. While the trto problems are "academic" (they can be reformulated as LP and solved much more efficiently), the other two problems have important engineering background.

Download the problems in SDPA format:

problem group	size (KB)
mater.zip	14,236
buck.zip	311
shmup.zip	863
trto.zip	151
vibra.zip	289

Description of the problems:

problem	n	m	objective
mater-1	103	222	-1.434654e+02
mater-2	423	1014	-1.415919e+02
mater-3	1439	3588	-1.339163e+02
mater-4	4807	12498	-1.342627e+02
mater-5	10143	26820	-1.338016e+02
mater-6	20463	56311	-1.335387e+02
trto1	36	25+36	1.1045 (exact value)
trto2	144	91+144	1.28 (exact value)
trto3	544	321+544	1.28 (exact value)
trto4	1200	673+1200	1.276582
trto5	3280	1761+3280	1.28 (exact value)
buck1	36	49+36	14.64192
buck2	144	193+144	292.3683
buck3	544	641+544	607.6055
buck4	1200	1345+1200	486.1421
buck5	3280	3521+3280	436.2390
vibra1	36	49+36	40.81901
vibra2	144	193+144	166.0153
vibra3	544	641+544	172.6130
vibra4	1200	1345+1200	165.6133
vibra5	3280	3521+3280	165.9029
shmup1	16	81+32	188.4148
shmup2	200	881+400	3462.427
shmup3	420	1801+840	2098.838
shmup4	800	3361+1600	7992.534
shmup5	1800	7441+3600	23858.867

Comments:

n is the number of variables, m the size of the matrix constraint 25+36 means: matrix constraint of size 25 and 36 linear constraints.

In the mater problems, the constraint matrix contains many small blocks. In all other problems, the constraint matrix contains two (in trto one) sparse diagonal blocks. The optimal objective values were computed by PENSDP and, in most cases, confirmed by SDPT3 and MOSEK. In the large-scale problems, the two/three codes may differ in the 5th-6th digit. In this case, we give the PENSDP value.

mater are problems of multiple-load free material optimization (area of structural optimization) modeled by linear SDP as described in [1]. All six examples solve the same problem (geometry, loads, boundary conditions) and differ only in the finite element discretization.

trto are problems from single-load truss topology design. Normally formulated as LP, here reformulated as SDP for testing purposes. (see, e.g., [2],[3])

vibra are single load truss topology problems with a vibration constraint. The constraint guarantees that the minimal self-vibration frequency of the optimal structure is bigger than a given value; see [4].

buck are single load truss topology problems with linearized global buckling constraint. Originally a nonlinear matrix inequality, the constraint should guarantee that the optimal structure is mechanically stable (it doesn't buckle); see [4].

shmup are minimum volume, single load free material optimization problems [5] with a vibration constraint and upper bound on the material density. The vibration constraint guarantees that the minimal self-vibration frequency of the optimal structure is bigger than a given value.

1. A. Ben-Tal, M. Kočvara, A. Nemirovski, and J. Zowe. Free material optimization via semidefinite programming: the multiload case with contact conditions. SIAM J. Optimization, 9(4): 813-832, 1999 DOI: 10.1137/S1052623497327994 and SIAM Review, 42(4): 695-715, 2000 DOI: 10.1137/S0036144500372081

2. A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM Philadelphia, 2001. DOI: 10.1137/1.9780898718829

3. M. Kočvara and J. Zowe. How mathematics can help in design of mechanical structures. In D.F. Griffiths and G.A. Watson, eds., Numerical Analysis 1995, Longman, Harlow, 1996, pp. 76-93. ISBN: 0 582 27633 0

4. M. Kočvara. On the modelling and solving of the truss design problem with global stability constraints. Structural and Multidisciplinary Optimization 23(3):189-203, 2002. DOI: 10.1007/s00158-002-0177-3

5. J. Zowe, M. Kočvara, and M. Bendsøe. Free Material Optimization via Mathematical Programming. Mathematical Programming, 79:445-466, 1997. DOI: 10.1007/BF02614328