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A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition…

In this paper, we propose an exact general algorithm for solving non-convex optimization problems, where the non-convexity arises due to the presence of an inverse S-shaped function. The proposed method involves iteratively approximating…

Constrained Optimization solution algorithms are restricted to point based solutions. In practice, single or multiple objectives must be satisfied, wherein both the objective function and constraints can be non-convex resulting in multiple…

We propose an optimization proxy in terms of iterative implicit gradient methods for solving constrained optimization problems with nonconvex loss functions. This framework can be applied to a broad range of machine learning settings,…

Many practical optimization problems lack strong convexity. Fortunately, recent studies have revealed that first-order algorithms also enjoy linear convergences under various weaker regularity conditions. While the relationship among…

In this paper we consider the problem of minimizing a convex function using a randomized block coordinate descent method. One of the key steps at each iteration of the algorithm is determining the update to a block of variables. Existing…

Optimizing a nonlinear function over nonconvex sets is challenging since solving convex relaxations may lead to substantial relaxation gaps and infeasible solutions that must be "rounded" to feasible ones, often with uncontrollable losses…

We unify nonlinear Farkas lemma and S-lemma to a generalized alternative theorem for nonlinear nonconvex system. It provides fruitful applications in globally solving nonconvex non-quadratic optimization problems via revealing the hidden…

We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum…

The problem of designing optimal quantization rules for sequential detectors is investigated. First, it is shown that this task can be solved within the general framework of active sequential detection. Using this approach, the optimal…

We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with…

The implicit convex feasibility problem attempts to find a point in the intersection of a finite family of convex sets, some of which are not explicitly determined but may vary. We develop simultaneous and sequential projection methods…

In this paper we consider non-smooth convex optimization problems with (possibly) infinite intersection of constraints. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets,…

Inspired by regularization techniques in statistics and machine learning, we study complementary composite minimization in the stochastic setting. This problem corresponds to the minimization of the sum of a (weakly) smooth function endowed…

This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the…

Constrained quasiconvex optimization problems appear in many fields, such as economics, engineering, and management science. In particular, fractional programming, which models ratio indicators such as the profit/cost ratio as fractional…

In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision…

We revisit the classical dual ascent algorithm for minimization of convex functionals in the presence of linear constraints, and give convergence results which apply even for non-convex functionals. We describe limit points in terms of the…

Optimization methods have been broadly applied to two classes of objects viz. (i) modeling and description of data and (ii) the determination of the stationary points of functions. Here, a theoretical basis is developed that optimizes an…

As science and engineering have become increasingly data-driven, the role of optimization has expanded to touch almost every stage of the data analysis pipeline, from signal and data acquisition to modeling and prediction. The optimization…