Exact SDP relaxations for quadratic programs with bipartite graph structures

For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. … Read more

Generalization of Doubly Nonnegative Cone: Focusing on Inner-Approximation for Generalized Copositive Cone

We aim to provide better relaxation for generalized completely positive (copositive) programming. We first develop an inner-approximation hierarchy for the generalized copositive cone over a symmetric cone. Exploiting this hierarchy as well as the existing hierarchy proposed by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, 2006), we then propose two (NN and ZVP) generalized doubly … Read more

Escaping Spurious Local Minima of Low-Rank Matrix Factorization Through Convex Lifting

This work proposes a rapid global solver for nonconvex low-rank matrix factorization (MF) problems that we name MF-Global. Through convex lifting steps, our method efficiently escapes saddle points and spurious local minima ubiquitous in noisy real-world data, and is guaranteed to always converge to the global optima. Moreover, the proposed approach adaptively adjusts the rank … Read more

Tight Probability Bounds with Pairwise Independence

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results towards finding tight probability bounds in this direction. Firstly, when $k = 1$, the tightest … Read more

A nearly linearly convergent first-order method for nonsmooth functions with quadratic growth

Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro’s decomposable … Read more

Error Bound and Isocost Imply Linear Convergence of DCA-based Algorithms to D-stationarity

We consider a class of structured nonsmooth difference-of-convex minimization, which can be written as the difference of two convex functions possibly nonsmooth with the second one in the format of the maximum of a finite convex smooth functions. We propose two extrapolation proximal difference-of-convex based algorithms for potential acceleration to converge to a weak/standard d-stationary … Read more

An implicit function formulation for optimization of discretized index-1 differential algebraic systems

A formulation for the optimization of index-1 differential algebraic equation systems (DAEs) that uses implicit functions to remove algebraic variables and equations from the optimization problem is described. The formulation uses the implicit function theorem to calculate derivatives of functions that remain in the optimization problem in terms of a reduced space of variables, allowing … Read more

An Exact Approach for Solving Pickup-and-Delivery Traveling Salesman Problems with Neighborhoods

This paper studies a variant of the traveling salesman problem called the pickup-and-delivery traveling salesman problem with neighborhoods that combines traditional pickup and delivery requirements with the flexibility of visiting the customers at locations within compact neighborhoods of arbitrary shape. We derive two optimality conditions for the problem, a local condition that verifies whether a … Read more

Duality assertions in vector optimization w.r.t. relatively solid convex cones in real linear spaces

We derive duality assertions for vector optimization problems in real linear spaces based on a scalarization using recent results concerning the concept of relative solidness for convex cones (i.e., convex cones with nonempty intrinsic cores). In our paper, we consider an abstract vector optimization problem with generalized inequality constraints and investigate Lagrangian type duality assertions … Read more

A Gauss-Newton-based Decomposition Algorithm for Nonlinear Mixed-Integer Optimal Control Problems

For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for the second algorithm stage that is … Read more