We investigate polyconvexity of the double well function $f(X) := |X-X_1|^2|X-X_2|^2$ for given matrices $X_1, X_2 \in \R^{n \times n}$. Such functions are fundamental in the modeling of phase transitions in materials, but their non-convex nature presents challenges for the analysis of variational problems. We prove that $f$ is polyconvex if and only if the … Read more
In this paper, we analyze augmented Lagrangian (AL) methods for mathematical programs with complementarity constraints (MPCCs), with emphasis on a variant that reformulates the complementarity constraints by slack variables and preserves them explicitly in the subproblems instead of penalizing them. Motivated by recent developments in nonlinear programming, we study quasi-normality-type constraint qualifications tailored to this … Read more
We propose a first order algorithm, a modified version of FISTA, to solve an optimization problem with an objective function that is the sum of a possibly nonconvex function, with Lipschitz continuous gradient, and a convex function which can be nonsmooth. The algorithm is shown to have an iteration complexity of \(\mathcal{O}(\epsilon^{-2})\) to find an … Read more
Contextual optimization enhances decision quality by leveraging side information to improve predictions of uncertain parameters. However, existing approaches face significant challenges when dealing with multimodal or mixtures of distributions. The inherent complexity of such structures often precludes an explicit functional relationship between the contextual information and the uncertain parameters, limiting the direct applicability of parametric … Read more
We study a generalization of the classical discrete-time, Linear-Quadratic-Gaussian (LQG) control problem where the noise distributions affecting the states and observations are unknown and chosen adversarially from divergence-based ambiguity sets centered around a known nominal distribution. For a finite horizon model with Gaussian nominal noise and a structural assumption on the divergence that is satisfied … Read more
This paper presents a restart version of the Adaptive Moment Estimation (Adam) method for bound constrained nonlinear optimization problems. It aims to avoid getting trapped in a local minima and enable exploration the global optimum. The proposed method combines an adapted restart strategy coupling with barrier methodology to handle the bound constraints. Computational comparison with … Read more
In this paper, we investigate the partial inverse knapsack problem, a bilevel optimization problem in which the follower solves a classical 0/1-knapsack problem with item profit values comprised of a fixed part and a modification determined by the leader. Specifically, the leader problem seeks a minimal change to given item profits such that there is … Read more
In Online Convex Optimization (OCO), when the stochastic gradient has a finite variance, many algorithms provably work and guarantee a sublinear regret. However, limited results are known if the gradient estimate has a heavy tail, i.e., the stochastic gradient only admits a finite \(\mathsf{p}\)-th central moment for some \(\mathsf{p}\in\left(1,2\right]\). Motivated by it, this work examines … Read more
Flow formulations have been widely studied for the one-dimensional cutting stock problem and several of its extensions. Among these, the so-called reflect model has shown the best empirical performance when solved directly with a general-purpose integer linear programming solver due to its reduced number of variables and constraints. However, existing adaptations of reflect for the … Read more
This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal smoothings is given. These provide if and only if characterizations of the set of optimal smoothings for … Read more