Modeling Cone Optimization Problems with COIN OS

Transcription

1 instance Modeling Cone Optimization s with COIN OS Joint work with Gus Gassman, Jun Ma, Kipp Martin Lehigh University Department of ISE INFORMS 2009, San Diego October 12, 2009

2 instance 1 Cone optimization Semidefinite optimization Special problems 2 instance Existing formats layout How to best represent the problems? 3 Design philosophy Declarations Data, functions 4

3 General cone optimization Cone optimization Semidefinite optimization Special problems instance min c T x Ax = b x K The cone K can be Linear: x 0 max b T y A T y + s = c s K Second-order: x 0 x 2 Rotated second-order: x 0 x 1 x 2:n, and x 0 0 Semidefinite: x is (can be assembled into) a symmetric, positive semidefinite matrix, or a product/intersection of these. robust control, combinatorics, polynomial and SOS, truss-topology, materials structure,...

4 Cone optimization Semidefinite optimization Special problems instance Semidefinite optimization Standard form min C X max b T y AX = b A y + S = C (P-D) X 0 S 0, where b, y R m, X, S, C R n2, A : R n2 R m Linear operator A AX = (A i X) m i=1 m A y = A i y i i=1 too restrictive

5 Cone optimization Semidefinite optimization Special problems instance Special forms Rank one, low rank A i A i = aa T, A i X = a T Xa can be exploited inside the IPM cannot be recovered exactly from A i General operators AX = AX + XA, or AX = AXB + BXA A is a large Kronecker product huge savings in storage and computation one needs to have A Cone intersections

6 instance Existing formats layout How to best represent the problems? Input formats What s out there SDP: SeDuMi, SDPT3, SDPpack, PENSDP, Sparse SDPA, extensions SOCP: MOSEK, LOQO, CPLEX CVX, Yalmip COIN-OS (first attempt) Common features based on the standard problem form not flexible hard to extend

7 instance Existing formats layout How to best represent the problems? layout x 1:2 x 3:7 0 mat(x 8:16 ) 0 C 1 C 2 C 3 A 11 A 12 A 13 = 3 A 21 A 22 A 23 I A 31 A 32 A 33 0 Declare variables and constraints Define the C j, A ij mappings and the RHS Very similar to LP The basic unit is different

8 instance Existing formats layout How to best represent the problems? A collection of cone optimization problems s/problem structures from robust optimization combinatorics stability and control polynomial optimization... Necessary language components a T Xa Tr(X) det(x) AXB + BXA X 1... Collection to be published later Joint work with Johan Löfberg and Michael C. Grant

9 instance Design philosophy Declarations Data, functions Current COIN OS conic constructs LP + cone constraints (our fault) very inefficient all the drawbacks of existing formats does not allow advanced operators Use matrix variables instead smallest unit further subdivision is artificial Use functions of matrices extend the OSnL library Goal: preprocessing

10 instance Design philosophy Declarations Data, functions Declarations Matrix variable from new/existing scalar variables verification is done here matrices can share variables Attributes symmetric, positive semidefinite Hermitian integer (MICLP!) matrix size bounds (interpreted according to the matrix type) Matrix parameters to be used in new functions det(m + X)

11 instance Design philosophy Declarations Data, functions Functions Create a library of matrix functions det(x) AX AXB + BXA λ min (X)... The arguments are matrices, not n 2 numbers! Verification is easier Extends the OSnL library

12 instance Conclusions We have a... collection of various cone problems list of constructs needed loose syntax We need an... exact syntax, documentation implementation into COIN OS XML-parser example library extensible preprocessing library OSsL extension