A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

Asymptotic Properties of Stationary Solutions of Coupled Nonconvex Nonsmooth Empirical Risk Minimization

This paper has two main goals: (a) establish several statistical properties—consistency, asymptotic distributions, and convergence rates—of stationary solutions and values of a class of coupled nonconvex and nonsmooth empirical risk-minimization problems and (b) validate these properties by a noisy amplitude-based phase-retrieval problem, the latter being of much topical interest. Derived from available data via sampling, these empirical risk-minimization problems are the computational workhorse of a population risk model that involves the minimization of an expected value of a random functional. When these minimization problems are nonconvex, the computation of their globally optimal solutions is elusive. Together with the fact that the expectation operator cannot be evaluated for general probability distributions, it becomes necessary to justify whether the stationary solutions of the empirical problems are practical approximations of the stationary solution of the population problem. When these two features, general distribution and nonconvexity, are coupled with nondifferentiability that often renders the problems “non-Clarke regular,” the task of the justification becomes challenging. Our work aims to address such a challenge within an algorithm-free setting. The resulting analysis is, therefore, different from much of the analysis in the recent literature that is based on local search algorithms. Furthermore, supplementing the classical global minimizer-centric analysis, our results offer a promising step to close the gap between computational optimization and asymptotic analysis of coupled, nonconvex, nonsmooth statistical estimation problems, expanding the former with statistical properties of the practically obtained solution and providing the latter with a more practical focus pertaining to computational tractability.

Globally Optimal Facility Locations for Continuous-Space Facility Location Problems

The continuous-space single- and multi-facility location problem has attracted much attention in previous studies. This study focuses on determining the globally optimal facility locations for two- and higher-dimensional continuous-space facility location problems when the Manhattan distance is considered. Before we propose the exact method, we start with the continuous-space single-facility location problem and obtain the global minimizer for the problem using a statistical approach. Then, an exact method is developed to determine the globally optimal solution for the two- and higher-dimensional continuous-space facility location problem, which is different from the previous clustering algorithms. Based on the newly investigated properties of the minimizer, we extend it to multi-facility problems and transfer the continuous-space facility location problem to the discrete-space location problem. To illustrate the effectiveness and efficiency of the proposed method, several instances from a benchmark are provided to compare the performances of different methods, which illustrates the superiority of the proposed exact method in the decision-making of the continuous-space facility location problems.

An Entropic Gradient Structure in the Network Dynamics of a Slime Mold

The approach to equilibrium in certain dynamical systems can be usefully described in terms of information-theoretic functionals. Well-studied models of this kind are Markov processes, chemical reaction networks, and replicator dynamics, for all of which it can be proven, under suitable assumptions, that the relative entropy (informational divergence) of the state of the system with respect to an equilibrium is nonincreasing over time. This work reviews another recent result of this type, which emerged in the study of the network optimization dynamics of an acellular slime mold, Physarum polycephalum. In this setting, not only the relative entropy of the state is nonincreasing, but its evolution over time is crucial to the stability of the entire system, and the equilibrium towards which the dynamics is attracted proves to be a global minimizer of the cost of the network.

The Moment-SOS hierarchy and the Christoffel-Darboux kernel

We consider the global minimization of a polynomial on a compact set B. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of L 2 (B,μ) where μ is an arbitrary reference measure whose support is exactly B. The resulting polynomial is a certain density (with respect to μ) of some signed measure on B. When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel-Darboux (CD) kernel and the Christoffel function associated with μ. In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).

Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

The critical points of the elastic energy among curves pinned at endpoints

<p style='text-indent:20px;'>In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.</p>

Consensus-based optimization on hypersurfaces: Well-posedness and mean-field limit

We introduce a new stochastic differential model for global optimization of nonconvex functions on compact hypersurfaces. The model is inspired by the stochastic Kuramoto–Vicsek system and belongs to the class of Consensus-Based Optimization methods. In fact, particles move on the hypersurface driven by a drift towards an instantaneous consensus point, computed as a convex combination of the particle locations weighted by the cost function according to Laplace’s principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In this paper, we study the well-posedness of the model and we derive rigorously its mean-field approximation for large particle limit.

Online Supervised Learning with Distributed Features over Multiagent System

Most current online distributed machine learning algorithms have been studied in a data-parallel architecture among agents in networks. We study online distributed machine learning from a different perspective, where the features about the same samples are observed by multiple agents that wish to collaborate but do not exchange the raw data with each other. We propose a distributed feature online gradient descent algorithm and prove that local solution converges to the global minimizer with a sublinear rate O 2 T . Our algorithm does not require exchange of the primal data or even the model parameters between agents. Firstly, we design an auxiliary variable, which implies the information of the global features, and estimate at each agent by dynamic consensus method. Then, local parameters are updated by online gradient descent method based on local data stream. Simulations illustrate the performance of the proposed algorithm.