Quasinormality is a classical constraint qualification originally introduced by Hestenes in 1975 and subsequently extensively studied in nonlinear programming and in problems with abstract constraints. In this paper, we extend this concept to the setting of nonlinear semidefinite programming (NSDP). We show that the proposed condition is strictly weaker than Robinson’s constraint qualification, while still … Read more
We focus on copositive and completely positive cones over symmetric cones of rank at least $5$, and in particular investigate whether these cones are spectrahedral shadows. We extend known results for nonnegative orthants of dimension at least $5$ to general symmetric cones of rank at least $5$. Specifically, we prove that when the rank of … Read more
Problem definition: Curb space has long been a scarce public resource in automobilized cities, serving competing uses for passenger parking and commercial activities. The rapid growth of e-commerce and home deliveries, combined with increasing urban density, has further intensified pressure on this already constrained resource, making effective curbspace management a critical policy challenge. Yet, in … Read more
We consider the oracle complexity of constrained convex optimization given access to a Linear Minimization Oracle (LMO) for the constraint set and a gradient oracle for the $L$-smooth, strongly convex objective. This model includes Frank-Wolfe methods and their many variants. Over the problem class of strongly convex constraint sets $S$, our main result proves that … Read more
Over the last decades, using piecewise-linear mixed-integer relaxations of nonlinear expressions has become a strong alternative to spatial branching for solving mixed-integer nonlinear programs. Since these relaxations give rise to large numbers of binary variables that encode interval selections, strengthening them is crucial. We investigate how to exploit the resulting combinatorial structure by integrating cutting-plane … Read more
Dantzig-Wolfe reformulation is a widely used technique for obtaining stronger relaxations in the context of branch-and-bound methods for solving integer optimization problems. Arc-Flow reformulations are a lesser known technique related to dynamic programming that has experienced a resurgence as result of the recent popularization of decision diagrams as a tool for solving integer programs. Although … Read more
Algorithms based on polyhedral outer approximations provide a powerful approach to solving mixed-integer nonlinear optimization problems. An initial relaxation of the feasible set is strengthened by iteratively adding linear inequalities and separating infeasible points. However, when the constraints are nonconvex, computing such separating hyperplanes becomes challenging. In this article, the moment-/sums-of-squares hierarchy is used in … Read more
In this paper, we consider an unconstrained composite convex optimisation problem. We propose an inertial forward–backward algorithm derived from an implicit– explicit discretisation of a second-order dynamical system with Hessian-driven damping. For α ≥ 3, we establish an O(1/d^2) convergence rate for the objective value gap. Furthermore, when α > 3, we prove that the … Read more
A stochastic gradient method for finite-sum minimization subject to deterministic linear constraints is proposed and analyzed. The procedure presented adapts the projected gradient method on a convex set to the use of both a stochastic gradient and a possibly inexact projection map. Under standard assumptions in the field of stochastic gradient methods, we provide theoretical … Read more
This paper studies a class of double-loop (inner-outer) algorithms for convex composite optimization. For unconstrained problems, we develop a restarted accelerated composite gradient method that attains the optimal first-order complexity in both the convex and strongly convex settings. For linearly constrained problems, we introduce inexact augmented Lagrangian methods, including a basic method and an outer-accelerated … Read more