Multi-Standard Quadratic Optimization: interior point methods and cone programming reformulation

Abstract

A Standard Quadratic Optimization Problem (StQP) consists of maximizing a (possibly indefinite) quadratic form over the standard simplex. Likewise, in a multi-StQP we have to maximize a (possibly indefinite) quadratic form over the Cartesian product of several standard simplices (of possibly different dimensions). Among many other applications, multi-StQPs occur in Machine Learning Problems. Several converging monotone interior point methods are established, which differ from the usual ones used in cone programming. Further, we prove an exact cone programming reformulation for establishing rigorous yet affordable bounds and finding improving directions.

Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

“…The latter problem arises from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics; see [7,8,9,11,17,18,21] and the references therein. A StBQP should also be not confused with a bi-StQP, which is a special case of a multi-StQP, a problem class studied recently in [6,19]. In bi-StQPs, the objective is a quadratic form, while the feasible set is a product of simplices, as in (1.1).…”

Graph-based quadratic optimization: A fast evolutionary approach

“…The results show that our dynamics is dramatically faster than standard approaches, while preserving the quality of the solution found. Future works include extending the proposed algorithm over a multi-population setting, with applications to multi-StQPs [14], as well as studying variations of the proposed dynamics.…”

Copositive optimization – Recent developments and applications

“…, 1] ⊤ ∈ R n : this subclass is also NP-hard (there can be up to ∼ 2 n /(1. 25 √ n) local nonglobal solutions [14]). Now, with E = ee ⊤ the n × n all-ones matrix, we have min x ⊤ Qx : x ∈ ∆ = min { Q, X : E, X = 1 , X ∈ C} .…”

“…But for a similar class arising in many applications, the Multi-Standard Quadratic Optimization Problems [25], dual attainability is not guaranteed while the duality gap is zero -an intermediate form between weak and strong duality [117]. A complete picture of possible attainability/duality gap constellations in primaldual pairs of copositive optimization problems is provided in [26], which also lists some elementary algebraic properties and counterexamples illustrating the difference between the semidefinite cone P and the copositive/completely positive cone C * /C.…”