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:

Description of the problems:

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. Bendsøe. Free Material Optimization via Mathematical Programming. Mathematical Programming, 79:445-466, 1997.