The complexity of simple models – a study of worst and typical hard cases for the Standard Quadratic Optimization Problem

In a Standard Quadratic Optimization Problem (StQP), a possibly indefinite quadratic form (the simplest nonlinear function) is extremized over the standard simplex, the simplest polytope. Despite this simplicity, the nonconvex instances of this problem class allow for remarkably rich patterns of coexisting local solutions, which are closely related to practical difficulties in solving StQPs globally. … Read more

New lower bounds and asymptotics for the cp-rank

Let $p_n$ denote the largest possible cp-rank of an $n\times n$ completely positive matrix. This matrix parameter has its significance both in theory and applications, as it sheds light on the geometry and structure of the solution set of hard optimization problems in their completely positive formulation. Known bounds for $p_n$ are $s_n=\binom{n+1}2-4$, the current … Read more

From seven to eleven: completely positive matrices with high cp-rank

We study $n\times n$ completely positive matrices $M$ on the boundary of the completely positive cone, namely those orthogonal to a copositive matrix $S$ which generates a quadratic form with finitely many zeroes in the standard simplex. Constructing particular instances of $S$, we are able to construct counterexamples to the famous Drew-Johnson-Loewy conjecture (1994) for … Read more

On the cp-rank and minimal cp factorizations of a completely positive matrix

We show that the maximal cp-rank of $n\times n$ completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of $n\times n$ completely positive matrices, thus answering a long standing question. We also show that the maximal cp-rank of $5\times 5$ matrices equals six, which proves the famous Drew-Johnson-Loewy conjecture … Read more

Think co(mpletely )positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone of all completely positive symmetric nxn matrices, and its … Read more

Improved bounds for interatomic distance in Morse clusters

We improve the best known lower bounds on the distance between two points of a Morse cluster in $\mathbb{R}^3$, with $\rho \in [4.967,15]$. Our method is a generalization of the one applied to the Lennard-Jones potential in~\cite{Schac}, and it also leads to improvements of lower bounds for the energy of a Morse cluster. Some of … Read more

Multi-Standard Quadratic Optimization Problems

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). Two converging monotone interior point methods are established. Further, we prove an … Read more

A First-Order Interior-Point Method for Linearly Constrained Smooth Optimization

We propose a first-order interior-point method for linearly constrained smooth optimization that unifies and extends first-order affine-scaling method and replicator dynamics method for standard quadratic programming. Global convergence and, in the case of quadratic programs, (sub)linear convergence rate and iterate convergence results are derived. Numerical experience on simplex constrained problems with 1000 variables is reported. … Read more

A conic duality Frank–Wolfe type theorem via exact penalization in quadratic optimization

The famous Frank–Wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a (possibly unbounded) polyhedron if the feasible values are bounded. This theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive-semidefiniteness or copositivity conditions, despite the fact that the … Read more

New results for molecular formation under pairwise potential minimization

We establish new lower bounds on the distance between two points of a minimum energy configuration of $N$ points in $\mathbb{R}^3$ interacting according to a pairwise potential function. For the Lennard-Jones case, this bound is 0.67985 (and 0.7633 in the planar case). A similar argument yields an estimate for the minimal distance in Morse clusters, … Read more