DEGENERACY NO MORE: A SURPERISINGLY SIMPLE METHOD
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Linear, Cone and Semidefinite Programming
A single loop method for quadratic minmax optimization
We consider a quadratic minmax problem with coupled inner constraints and propose a method to compute a class of stationary points. To motivate the need to compute such stationary points, we first show that they are meaningful, in the sense that they can be locally optimal for our problem under suitable linear independence and second-order … Read more
Solving Stengle’s Example in Rational Arithmetic: Exact Values of the Moment-SOS Relaxations
We revisit Stengle’s classical univariate polynomial optimization example $\min 1-x^2$ s.t. $(1-x^2)^3\ge 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r\ge 3$ has the exact value $-1/r(r-2)$. For this we construct in rational arithmetic a dual polynomial sum-of-squares (SOS) certificate and a primal moment sequence … Read more
Global optimization of low-rank polynomials
This work considers polynomial optimization problems where the objective admits a low-rank canonical polyadic tensor decomposition. We introduce LRPOP (low-rank polynomial optimization), a new hierarchy of semidefinite programming relaxations for which the size of the semidefinite blocks is determined by the canonical polyadic rank rather than the number of variables. As a result, LRPOP can … Read more
Semidefinite hierarchies for diagonal unitary invariant bipartite quantum states
We investigate questions about the cone \(\mathrm{SEP}_n\) of separable bipartite states, consisting of the Hermitian matrices acting on \(\mathbb{C}^n\otimes\mathbb{C}^n\) that can be written as conic combinations of rank one matrices of the form \(xx^*\otimes yy^*\) with \(x,y\in\mathbb{C}^n\). Bipartite states that are not separable are said to be entangled. Detecting quantum entanglement is a fundamental task … Read more
Density, Determinacy, Duality and a Regularized Moment-SOS Hierarchy
The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar’s Positivstellensatz, which requires the feasible set to satisfy the algebraic Archimedean property. In this paper, we introduce a regularized moment-SOS hierarchy capable of handling problems on unbounded sets or bounded sets violating the Archimedean … Read more
A Dual Riemannian ADMM Algorithm for Low-Rank SDPs with Unit Diagonal
This paper proposes a dual Riemannian alternating direction method of multipliers (ADMM) for solving low-rank semidefinite programs with unit diagonal constraints. We recast the ADMM subproblem as a Riemannian optimization problem over the oblique manifold by performing the Burer-Monteiro factorization. Global convergence of the algorithm is established assuming that the subproblem is solved to certain … Read more
Semidefinite programming via Projective Cutting Planes for dense (easily-feasible) instances
The cone of positive semi-definite (SDP) matrices can be described by an infinite number of linear constraints. It is well-known that one can optimize over such a feasible area by standard Cutting Planes, but work on this idea remains a rare sight, likely due to its limited practical appeal compared to Interior Point Methods (IPMs). … Read more
Sparse Multiple Kernel Learning: Alternating Best Response and Semidefinite Relaxations
We study Sparse Multiple Kernel Learning (SMKL), which is the problem of selecting a sparse convex combination of prespecified kernels for support vector binary classification. Unlike prevailing \(\ell_1\)‐regularized approaches that approximate a sparsifying penalty, we formulate the problem by imposing an explicit cardinality constraint on the kernel weights and add an \(\ell_2\) penalty for robustness. … Read more
The SCIP Optimization Suite 10.0
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming (CIP) framework SCIP. This report discusses the enhancements and extensions included in SCIP Optimization Suite 10.0. The updates in SCIP 10.0 include a new solving mode for exactly solving rational mixed-integer linear programs, a new presolver … Read more