We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems (QOPs) to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to … Read more
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of matchings, extended by a binary number indicating whether the matching contains two specific edges. This polytope is associated to the quadratic matching problem with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein … Read more
We present a new derivative-free trust-region (DFTR) algorithm to solve general nonlinear constrained problems with the use of an augmented Lagrangian method. No derivatives are used, neither for the objective function nor for the constraints. An augmented Lagrangian method, known as an effective tool to solve equality and inequality constrained optimization problems with derivatives, is … Read more
In this paper we study stochastic quasi-Newton methods for nonconvex stochastic optimization, where we assume that noisy information about the gradients of the objective function is available via a stochastic first-order oracle ($\SFO$). We propose a general framework for such methods, for which we prove almost sure convergence to stationary points and analyze its worst-case … Read more
In this paper we consider the composite self-concordant (CSC) minimization problem, which minimizes the sum of a self-concordant function $f$ and a (possibly nonsmooth) proper closed convex function $g$. The CSC minimization is the cornerstone of the path-following interior point methods for solving a broad class of convex optimization problems. It has also found numerous … Read more
Recently, the 1-bit compressive sensing (1-bit CS) has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. In this paper, we first … Read more
A small improvement in the structure of the material could save the manufactory a lot of money. The free material design can be formulated as an optimization problem. However, due to its large scale, second-order methods cannot solve the free material design problem in reasonable size. We formulate the free material optimization (FMO) problem into … Read more
The Carathéodory number k(K) of a pointed closed convex cone K is the minimum among all the k for which every element of K can be written as a nonnegative linear combination of at most k elements belonging to extreme rays. Carathéodory’s Theorem gives the bound k(K) <= dim (K). In this work we observe … Read more
Composite optimization models consist of the minimization of the sum of a smooth (not necessarily convex) function and a non-smooth convex function. Such models arise in many applications where, in addition to the composite nature of the objective function, a hierarchy of models is readily available. It is common to take advantage of this hierarchy … Read more
Detailed modeling of gas transport problems leads to nonlinear and nonconvex mixed-integer optimization or feasibility models (MINLPs) because both the incorporation of discrete controls of the network as well as accurate physical and technical modeling is required in order to achieve practical solutions. Hence, ignoring certain parts of the physics model is not valid for … Read more