Abstract
Solving the large‐scale problems with semidefinite programming (SDP) constraints is of great importance in modeling and model reduction of complex system, dynamical system, optimal control, computer vision, and machine learning. However, existing SDP solvers are of large complexities and thus unavailable to deal with large‐scale problems. In this paper, we solve SDP using matrix generation, which is an extension of the classical column generation. The exponentiated gradient algorithm is also used to solve the special structure subproblem of matrix generation. The numerical experiments show that our approach is efficient and scales very well with the problem dimension. Furthermore, the proposed algorithm is applied for a clustering problem. The experimental results on real datasets imply that the proposed approach outperforms the traditional interior‐point SDP solvers in terms of efficiency and scalability.
Highlights
Semidefinite programming SDP is a technique widely used in modeling of complex systems and some important issues in computer vision and machine learning
Ij Xi2j is the Frobenius norm, K is a n × n symmetric p.s.d. matrix, and k is a positive integer. This problem is important in the issue of the affinity matrix normalization of spectral clustering 17
We compare the proposed algorithm with the convex optimization solver, namely, SDPT3 18 which is used as internal solvers in the disciplined convex programming software CVX 19, 20
Summary
Semidefinite programming SDP is a technique widely used in modeling of complex systems and some important issues in computer vision and machine learning. Examples of application include model reduction 1 , modeling of nonlinear systems 2, 3 , optimal control 4 , clustering 5–7 , robust Euclidean embedding 8 , kernel matrix learning 9 , and metric learning 10. It can be seen in 5, 7 that SDP relaxation can produce more accurate estimates than spectral methods. We propose the matrix generation-based iteration approach to solve general SDP optimization problems. The method proposed here can be seen as an extension of column generation to solve SDP problems. We generalize EG in the sense that the proposed method solves optimization with a semidefinite matrix whose eigenvalues are on a simplex. I a bold lower-case letter x : a column vector, ii an upper-case letter X : a matrix, iii A 0: a positive semidefinite p.s.d. matrix, iv a b: the component-wise inequality between two vectors, v Êm×n : the vector space of real matrices of size m × n, vi Ë: the space of real matrices, vii Ën: the space of symmetric matrices of size n × n, viii Ën: the space of symmetric positive semidefinite matrices of size n × n, ix A, B Tr A B : the inner product defined on the above spaces, x Tr · : the trace of a matrix, xi Δn : x ∈ Ên | x 0, 1 x 1: the n-dimensional simplex set for vectors, xii Pn : {X ∈ Ën | X 0, Tr X 1}: the density matrix set, xiii Qn : {X ∈ Ën | X 0, Tr X 1, rank X 1}: the dyad set of matrices
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