The Maximum Singularity Degree for Linear and Semidefinite Programming
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Semi-definite Programming
T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization
We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng, Huang and Hu in 2022. In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into … Read more
Cuts and semidefinite liftings for the complex cut polytope
We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices \(xx^{\mathrm{H}}\), where the elements of \(x \in \mathbb{C}^n\) are \(m\)th unit roots. These polytopes have applications in \({\text{MAX-3-CUT}}\), digital communication technology, angular synchronization and more generally, complex quadratic programming. For \({m=2}\), the complex cut polytope corresponds to the well-known cut polytope. … Read more
Refined TSSOS
The moment-sum of squares hierarchy by Lasserre has become an established technique for solving polynomial optimization problems. It provides a monotonically increasing series of tight bounds, but has well-known scalability limitations. For structured optimization problems, the term-sparsity SOS (TSSOS) approach scales much better due to block-diagonal matrices, obtained by completing the connected components of adjacency … Read more
Sparse Polynomial Optimization with Unbounded Sets
This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the running intersection property, we prove that this hierarchy … Read more
Post-Processing with Projection and Rescaling Algorithms for Semidefinite Programming
We propose the algorithm that solves the symmetric cone programs (SCPs) by iteratively calling the projection and rescaling methods the algorithms for solving exceptional cases of SCP. Although our algorithm can solve SCPs by itself, we propose it intending to use it as a post-processing step for interior point methods since it can solve the … Read more
Solving moment and polynomial optimization problems on Sobolev spaces
Using standard tools of harmonic analysis, we state and solve the problem of moments for positive measures supported on the unit ball of a Sobolev space of multivariate periodic trigonometric functions. We describe outer and inner semidefinite approximations of the cone of Sobolev moments. They are the basic components of an infinite-dimensional moment-sums of squares … Read more
A semi-smooth Newton method for general projection equations applied to the nearest correlation matrix problem
In this paper, we extend and investigate the properties of the semi-smooth Newton method when applied to a general projection equation in finite dimensional spaces. We first present results concerning Clarke’s generalized Jacobian of the projection onto a closed and convex cone. We then describe the iterative process for the general cone case and establish … Read more
Strong global convergence properties of algorithms for nonlinear symmetric cone programming
Sequential optimality conditions have played a major role in proving strong global convergence properties of numerical algorithms for many classes of optimization problems. In particular, the way complementarity is dealt is fundamental to achieve a strong condition. Typically, one uses the inner product structure to measure complementarity, which gives a very general approach to a … Read more
On Rank-Monotone Graph Operations and Minimal Obstruction Graphs for the Lovász-Schrijver SDP Hierarchy
We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász-Schrijver SDP operator LS_+, with a particular focus on finding and characterizing the smallest graphs with a given LS_+-rank (the least number of iterations of the LS_+ operator on the fractional stable set polytope to compute the stable set … Read more