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 |
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.
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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
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A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM Philadelphia, 2001. DOI: 10.1137/1.9780898718829
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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
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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
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J. Zowe, M. Kočvara, and M. Bendsøe. Free Material Optimization via Mathematical Programming. Mathematical Programming, 79:445-466, 1997. DOI: 10.1007/BF02614328