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This work develops an algorithm for PDE-constrained shape optimization based on Lipschitz transformations. Building on previous work in this field, the $p$-Laplace operator is utilized to approximate a descent method for Lipschitz shapes.…
First-order optimization methods are crucial for solving large-scale data processing problems, particularly those involving convex non-smooth composite objectives. For such problems with convex non-smooth composite objectives, we introduce…
This article considers a box-constrained global optimization problem for Lipschitz-continuous functions with an unknown Lipschitz constant. Motivated by the famous DIRECT (DIviding RECTangles), a new HALRECT (HALving RECTangles) algorithm…
In this paper, a new theory is developed for first-order stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal…
This paper focuses on optimization problems constrained by Parametric Variational Inequalities (PVI) defined on a moving set. Unlike most existing works on mathematical programs with equilibrium constraints, the equilibrium constraints have…
Bilevel optimization (BLO) problem, where two optimization problems (referred to as upper- and lower-level problems) are coupled hierarchically, has wide applications in areas such as machine learning and operations research. Recently, many…
We give an algorithm to compute a one-dimensional shape-constrained function that best fits given data in weighted-$L_{\infty}$ norm. We give a single algorithm that works for a variety of commonly studied shape constraints including…
The work is devoted to the construction of efficient and applicable to real tasks first-order methods of convex optimization, that is, using only values of the target function and its derivatives. Construction uses OGM-G, fast gradient…
In this work, we propose an efficient minimax optimal global optimization algorithm for multivariate Lipschitz continuous functions. To evaluate the performance of our approach, we utilize the average regret instead of the traditional…
It is well known that both gradient descent and stochastic coordinate descent achieve a global convergence rate of $O(1/k)$ in the objective value, when applied to a scheme for minimizing a Lipschitz-continuously differentiable,…
We consider a composite optimization problem where the sum of a continuously differentiable and a merely lower semicontinuous function has to be minimized. The proximal gradient algorithm is the classical method for solving such a problem…
This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a…
Hidden convex optimization is such a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we focus on checking local optimality in hidden…
We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. We prove convergence of our numerical method under reasonably mild assumptions: boundedness and…
We study a generic class of decentralized algorithms in which $N$ agents jointly optimize the non-convex objective $f(u):=1/N\sum_{i=1}^{N}f_i(u)$, while only communicating with their neighbors. This class of problems has become popular in…
We introduce a novel approach addressing global analysis of a difficult class of nonconvex-nonsmooth optimization problems within the important framework of Lagrangian-based methods. This genuine nonlinear class captures many problems in…
Constrained Optimum Path (COP) problems appear in many real-life applications, especially on communication networks. Some of these problems have been considered and solved by specific techniques which are usually difficult to extend. In…
Algorithms for online learning typically require one or more boundedness assumptions: that the domain is bounded, that the losses are Lipschitz, or both. In this paper, we develop a new setting for online learning with unbounded domains and…
We propose a new gradient descent algorithm with added stochastic terms for finding the global optimizers of nonconvex optimization problems. A key component in the algorithm is the adaptive tuning of the randomness based on the value of…
Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus.…