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In this paper we study a problem of looking for an optimal solution of a system of the differential equations with a control and an optimized function. The system of differential equations is changed for two systems with the upper and lower…
In this paper we present two frameworks in which global maximization of a bounded hessian function over a strongly convex set can be reduced to convex optimization. The first presented framework is a continuation of one of our previous…
In Speakman and Lee (2017), we analytically developed the idea of using volume as a measure for comparing relaxations in the context of spatial branch-and-bound. Specifically, for trilinear monomials, we analytically compared the three…
We develop a spatial branch-and-cut approach for nonconvex Quadratically Constrained Quadratic Programs with bounded complex variables (CQCQP). Linear valid inequalities are added at each node of the search tree to strengthen semidefinite…
In this paper we study the problem of calculating the convex hull of certain affine algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call…
Let X be a polyhedral complex with finitely many isometry classes of links. We establish a restriction on the covolumes of uniform lattices acting on X. When X is two-dimensional and has all links isometric to either a complete bipartite…
We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem…
We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the $p$-norm function, where $p$…
We establish new upper and lower bounds on the number of queries required to test convexity of functions over various discrete domains. 1. We provide a simplified version of the non-adaptive convexity tester on the line. We re-prove the…
In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the "\emph{polynomials} plus \emph{bubbles}" form. The nonlinear problem is…
We consider scalar equilibrium problems governed by a bifunction in a finite-dimensional framework. By using classical arguments in Convex Analysis, we show that under suitable generalized convexity assumptions imposed on the bifunction,…
Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue \lambda. We give upper and lower bounds on the inner radius of the type C/\lambda^k. Our proof is based on a local behavior of…
We investigate a mixed 0-1 conic quadratic optimization problem with indicator variables arising in mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed charges or cardinality constraints.…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products…
We introduce two notions of convexity for an infinite regular tree. For these two notions we show that given a continuous boundary datum there exists a unique convex envelope on the tree and characterize the equation that this envelope…
Abstract Equivalent conditions that make the convex subdifferential maximal monotone are investigated in the general settings of locally convex spaces.
The convex hull of N independent random points chosen on the boundary of a simple polytope in R^n is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are…
An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the…
The maximum-entropy sampling problem is a fundamental and challenging combinatorial-optimization problem, with application in spatial statistics. It asks to find a maximum-determinant order-$s$ principal submatrix of an order-$n$ covariance…