Sparse SDP problems

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 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 544 312+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, eg, [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. Kocvara, 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 and SIAM Review, 42(4): 695-715, 2000.
  2. A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM Philadelphia, 2001.
  3. M. Kocvara 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.
  4. M. Kocvara. On the modelling and solving of the truss design problem with global stability constraints. Structural and Multidisciplinary Optimization 23(3):189-203, 2002.
  5. J. Zowe, M. Kocvara and M. Bends√łe. Free Material Optimization via Mathematical Programming. Mathematical Programming, 79:445-466, 1997.

Michal Kocvara
February 2, 2017