In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many app...

Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed find asymptotic results for very large graphs, so it seems that the not suitable finding numbers. But this intuition wrong, and we will develop technique do just in paper. We new upper bounds many graph hypergraph As result, prove values $R(K_4^-,K_4^-,K_4^-)=28$, $R(...

Random projections can reduce the dimensionality of point sets while keeping approximate congruence. Applying random to optimization problems raises many theoretical and computational issues. Most issues in application conic programming were addressed Liberti et al. (Linear Algebr. Appl. 626:204–220, 2021) [1]. This paper focuses on semidefinite programming.

Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximum-satisf...

Interior point methods can be extended to a number of cones (self-dual homogeneous cones) • Rn (linear programming) • vectorized symmetric matrices over real numbers (semidefinite programming) • vectorized Hermitian matrices over complex numbers • vectorized Hermitian matrices over quaternions • vectorized Hermitian 3×3 matrices over octonions Grötschel, Lovász and Schrijver [3]: semidefinite p...

In previous lectures, we discussed a few fairly direct connections between quantum information theoretic notions and semidefinite programs. For instance, the semidefinite program associated with an optimization over measurements is quite simple and direct, and if you are familiar with the Choi representation of channels the same can be said about our semidefinite program for optimizing over cha...