As a starting point of our research, we show that, for a fixed order $\gamma \ge 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of ...$$$$$$$\mathrm{}$$\mathrm{}$

We develop a fully asynchronous proximal bundle method for solving non-smooth, convex optimization problems. The algorithm can be used as a drop-in replacement for classic bundle methods, i.e., the function must be given by a first-order oracle ...

Several classical adaptive optimization algorithms, such as line search and trust-region methods, have been recently extended to stochastic settings where function values, gradients, and Hessians in some cases, are estimated via stochastic ...

We devise a method for minimizing a low-rank quasiconcave objective function over a polytope by first projecting the polytope’s normal fan, then using the projected fan to obtain candidate solutions. When the polytope’s maximal number of ...

Scenario optimization is an approach to data-driven decision-making that has been introduced some fifteen years ago and has ever since then grown fast. Its most remarkable feature is that it blends the heuristic nature of data-driven methods with ...

We consider the global minimization of smooth functions based solely on function evaluations. Algorithms that achieve the optimal number of function evaluations for a given precision level typically rely on explicitly constructing an approximation ...${}^{\mathrm{}}$${}^{\mathrm{}}$${}^{\mathrm{}\mathrm{}\mathrm{}}$

Risk measures are commonly used to capture the risk preferences of decision-makers (DMs). The decisions of DMs can be nudged or manipulated when their risk preferences are influenced by factors such as the availability of information about the ...

This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies ...

This paper introduces a new storage-optimal first-order method, CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs, is characterized by low-rank ...

We consider min–max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the ...

This paper settles an open and challenging question pertaining to the design of simple and optimal high-order methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding $x^{\star }\in \mathcal{X}$ such that $⟨F(x),x-x^{\star }⟩\ge 0$ for all $x\in \mathcal{X}$. ...${\mathbb{}}^{}{\mathbb{}}^{}$${\mathrm{}}^{}$$\mathrm{}$${}^{\mathrm{}}$${}^{\mathrm{}\mathrm{}}\mathrm{}$${}^{\mathrm{}\mathrm{}}\mathrm{}$${}^{}$${}^{\mathrm{}\mathrm{}}$${}^{}$$\mathrm{}{}^{\mathrm{}\mathrm{}}$${}^{}$${}^{}$${}^{}$${}^{}$${}^{\mathrm{}}$${}^{}$

In this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study ...

We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are ...

Given a fixed finite metric space $(V,\mu )$, the minimum 0-extension problem, denoted as $\mathtt{0}\mathtt{\text{-}}\mathtt{Ext}[\mu ]$, is equivalent to the following optimization problem: minimize function of the form ${min}_{x\in V^n}{\sum }_i{f}_i(x_i)+{\sum }_{\mathit{ij}}{c}_{\mathit{ij}}\mu (x_i,x_j)$ where $f_i:V\to \mathbb{R}$ are functions given by $f_i(...{}_{}{}_{}{}_{\mathit{}}{}_{}$${}_{\mathit{}}{}_{\mathit{}}$$\mathtt{}\mathtt{}$$$$\mathtt{}\mathtt{}$$\mathtt{}\mathtt{}$${}^{}$${}_{}{}_{}$${}_{}{}_{}$$$$\left(\begin{array}{c}\\ \mathrm{}\end{array}\right)$$$$\mathtt{}\mathtt{}$

We study linear programming relaxations of nonconvex quadratic programs given by the reformulation–linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible ...

We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts ...

A key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new ...${}^{\mathrm{}}$${}^{\mathrm{}\mathrm{}}$

We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms ...

We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider (i) classes of optimization problems of ...

We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the ...${\mathbb{}}^{}$${\mathbb{}}^{\mathrm{}}$${}^{}$