The existence of Lagrange multipliers at a solution of a nonlinear optimization problem constitutes one of the cornerstones of modern optimization theory, with many important consequences for guiding algorithmic procedures towards a solution, defining stopping criteria, performing stability analysis, and several other aspects. However, the proof of this result is often intricate, relying on non-trivial … Read more
Covering properties build the foundation of stability and sensitivity analysis of solutions to a generalized equation and more specific optimization-related stationarity and equilibrium problems. It has been well-understood that metric regularity of the mapping defining the generalized equation is a key to furnish Lipschitzian stability of the solution of interest. With this work, we want … Read more
The boosted Frank-Wolfe algorithm accelerates the classical Frank-Wolfe algorithm by better aligning the update direction with the negative gradient. Its analysis, however, has been limited to deterministic convex problems, with step sizes that require either line search or knowledge of the Lipschitz constant of the gradient. We develop a novel step size strategy that does … Read more
We study a class of constrained nonconvex-nonconcave minimax optimization problems in which the inner maximization involves potentially complex constraints. Under the assumption that the inner problem of a novel lifted minimax reformulation satisfies a local Kurdyka-Łojasiewicz (KL) condition, we show that the maximal function of the original problem enjoys a local generalized Hölder smoothness property. … Read more
Direct search is one of the most popular derivative-free optimization paradigms, that relies on exploring the variable space using polling directions. To analyze and implement direct search, one typically relies on positive spanning sets. This concept is somewhat decorrelated from interpolation-based sets used in model-based algorithms, another class of derivative-free optimization methods. This discrepancy is … Read more
In this study, we model the supervised feature selection problem using a novel approach: convex bi-objective optimization. Traditional methods have addressed this problem by maximizing relevance to class labels and minimizing redundancy among features. Recently, Wang et al. [30] formulated this problem as a single-objective convex optimization, yielding only a unique solution. Unlike that, we … Read more
In this work we develop and analyze a semi-smooth Newton method for the general non- linear conic programming problem. In particular, we study the problem with a generalized simplicial cone, i.e., the image of a symmetric cone under a linear mapping. We generalize Robinson’s normal equations to a conic setting, yielding what we call the … Read more
In this paper, we consider nonlinear optimization problems with a stochastic objective function and deterministic equality constraints. We propose an inexact two-stepsize stochastic sequential quadratic programming (SQP) algorithm and analyze its worst-case complexity under mild assumptions. The method utilizes a step decomposition strategy and handles stochastic gradient estimates by assigning different stepsizes to different components … Read more
An algorithm is proposed for solving optimization problems with stochastic objective and deterministic equality and inequality constraints. This algorithm is objective-function-free in the sense that it only uses the objective’s gradient and never evaluates the function value. It is based on an adaptive selection of function-decreasing and constraint-improving iterations, the first ones using an Adagrad-type … Read more
In this paper, we propose a descent method for composite optimization problems with linear operators. Specifically, we first design a structure-exploiting preconditioner tailored to the linear operator so that the resulting preconditioned proximal subproblem admits a closed-form solution through its dual formulation. However, such a structure-driven preconditioner may be poorly aligned with the local curvature … Read more