We give a characterization of the Hoffman constant of a system of linear constraints in $\R^n$ {\em relative} to a {\em reference polyhedron} $R\subseteq\R^n$. The reference polyhedron $R$ represents constraints that are easy to satisfy such as box constraints. In the special case $R = \R^n$, we obtain a novel characterization of the classical Hoffman constant.
More precisely, suppose $R\subseteq \mathbb{R}^n$ is a reference polyhedron, $A\in \R^{m\times n},$ and $A(R):=\{Ax: x\in R\}$. We characterize the sharpest constant $H(A|R)$ such that for all $b \in A(R) + \R^m_+$ and $u\in R$ \[ \dist(u, P_{A}(b)\cap R) \le H(A|R) \cdot \|(Au-b)_+\|, \] where $P_A(b) = \{x\in \R^n:Ax\le b\}$. Our characterization is stated in terms of the largest of a canonical collection of easily computable Hoffman constants. Our characterization in turn suggests new algorithmic procedures to compute Hoffman constants.
A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance between the two players is two. We show that the symmetric rendezvous value is within the interval (4.1520, 4.2574), which considerably improves the previous best-known result (3.9546, 4.3931). To achieve the improved bounds, we call upon results from absorbing Markov chain theory and mathematical programming theory—particularly fractional quadratic programming and semidefinite programming. Moreover, we also establish some important properties of this problem, which could be of independent interest and useful for resolving this problem completely. Finally, we conjecture that the symmetric rendezvous value is asymptotically equal to 4.25 based on our numerical calculations.
Suppose $A\in \mathbb{R}^{m\times n}$ and consider the following canonical systems of inequalities defined by $A$: $$ \begin{array}{l} Ax=b\\ x \ge 0 \end{array} \qquad \text{ and }\qquad A^T y - c \le 0. $$ We establish some novel duality relationships between the Hoffman constants for the above constraint systems of linear inequalities provided some suitable Slater condition holds. The crux of our approach is a Hoffman duality inequality for polyhedral systems of constraints. The latter in turn yields an interesting duality identity between the Hoffman constants of the following box-constrained systems of inequalities: $$ \begin{array}{l} Ax=b\\ \ell \le x \le u \end{array}\qquad \text{ and }\qquad \ell \le A^T y - c \le u $$ for $\ell, u\in \mathbb{R}^n$ with $\ell < u.$
In this work, we present a new characterization of symmetric $H^+$-tensors. It is known that a symmetric tensor is an $H^+$-tensor if and only if it is a generalized diagonally dominant tensor with nonnegative diagonal elements. By exploring the diagonal dominance property, we derive new necessary and sufficient conditions for a symmetric tensor to be an $H^+$-tensor. Based on these conditions, we propose a new method that allows to check if a tensor is a symmetric $H^+$-tensor in polynomial time. In particular, this allows to efficiently compute the minimum $H$-eigenvalue of tensors in the related and important class of $M$-tensors. Furthermore, we show how this result can be used to approximately solve polynomial optimization problems.
A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max $k$-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max $k$-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max $k$-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.
We show the equivalence among the following three condition measures of a full column rank matrix $A$: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of $A$. Our results provide a procedure to estimate $\chi(A)$ thereby opening an avenue to identify classes of linear programs solvable in polynomial time in the real model of computation.
A certificate of non-negativity is a way to write a given function so that its non-negativity becomes evident. Certificates of non-negativity are fundamental tools in optimization, and they underlie powerful algorithmic techniques for various types of optimization problems. We propose certificates of non-negativity of polynomials based on copositive polynomials. The certificates we obtain are valid for generic semialgebraic sets and have a fixed small degree, while commonly used sums-of-squares (SOS) certificates are guaranteed to be valid only for compact semialgebraic sets and could have large degree. Optimization over the cone of copositive polynomials is not tractable, but this cone has been well studied. The main benefit of our copositive certificates of non-negativity is their ability to translate results known exclusively for copositive polynomials to more general semialgebraic sets. In particular, we show how to use copositive polynomials to construct structured (e.g., sparse) certificates of non-negativity, even for unstructured semialgebraic sets. Last but not least, copositive certificates can be used to obtain not only hierarchies of tractable lower bounds, but also hierarchies of tractable upper bounds for polynomial optimization problems.
Microgrid formation is a potential solution in post-disaster electric grid recovery efforts. Recent works propose distribution level microgrid formation models using mixed-integer linear programming techniques. However, these models can only be solved for small and medium size power systems due to their computational intractability. In this paper, we introduce a heuristic approach to approximately solve the post-disturbance microgrid formation problem for medium to large, more realistic, instances. Furthermore, the proposed approach allows to approximately solve the pre-disturbance microgrid formation problem (a stochastic version of the post-disturbance problem), in which the aim is to allocate extra generation capacity to the network to immunize it, as best as possible, against an uncertain cascading failure. Our results are illustrated by solving versions of the problem on ten test cases with up to 3012 buses.
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