Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization

We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable $O(1/T)$ convergence guarantees in terms of both their classic primal gap and a … Read more

Data-Driven Reliable Facility Location Design

We study the reliable (uncapacitated) facility location (RFL) problem in a data-driven environment where historical observations of random demands and disruptions are available. Owing to the combinatorial optimization nature of the RFL problem and the mixed-binary randomness of parameters therein, the state-of-the-art RFL models applied to the data-driven setting either suggest overly conservative solutions, or … Read more

Krasnoselskii-Mann Iterations: Inertia, Perturbations and Approximation

This paper is concerned with the study of a family of fixed point iterations combining relaxation with different inertial (acceleration) principles. We provide a systematic, unified and insightful analysis of the hypotheses that ensure their weak, strong and linear convergence, either matching or improving previous results obtained by analysing particular cases separately. We also show … Read more

Exploring Nonlinear Distance Metrics for Lipschitz Constant Estimation in Lower Bound Construction for Global Optimization

Bounds play a crucial role in guiding optimization algorithms, improving their speed and quality, and providing optimality gaps. While Lipschitz constant-based lower bound construction is an effective technique, the quality of the linear bounds depends on the function’s topological properties. In this research, we improve upon this by incorporating nonlinear distance metrics and surrogate approximations … Read more

Refined TSSOS

The moment-sum of squares hierarchy by Lasserre has become an established technique for solving polynomial optimization problems. It provides a monotonically increasing series of tight bounds, but has well-known scalability limitations. For structured optimization problems, the term-sparsity SOS (TSSOS) approach scales much better due to block-diagonal matrices, obtained by completing the connected components of adjacency … Read more

Policy with guaranteed risk-adjusted performance for multistage stochastic linear problems

Risk-averse multi-stage problems and their applications are gaining interest in various fields of applications. Under convexity assumptions, the resolution of these problems can be done with trajectory following dynamic programming algorithms like Stochastic Dual Dynamic Programming (SDDP) to access a deterministic lower bound, and dual SDDP for deterministic upper bounds. In this paper, we leverage … Read more

Optimization Over Trained Neural Networks: Taking a Relaxing Walk

Besides training, mathematical optimization is also used in deep learning to model and solve formulations over trained neural networks for purposes such as verification, compression, and optimization with learned constraints. However, solving these formulations soon becomes difficult as the network size grows due to the weak linear relaxation and dense constraint matrix. We have seen … Read more

Sparse Polynomial Optimization with Unbounded Sets

This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the running intersection property, we prove that this hierarchy … Read more

Active Set-based Inexact Proximal Bundle Algorithm for Stochastic Quadratic Programming

In this paper, we examine two-stage stochastic quadratic programming problems, where the objective function of the first and second stages are quadratic functions, and the constraints are linear. The uncertainty is associated with the second-stage right-hand side and variable bounds. In large-scale settings, when the number of scenarios necessary to represent the underlying stochastic process … Read more

Tactical workforce sizing and scheduling decisions for last-mile delivery

We tackle the problems of workforce sizing and shift scheduling of a logistic operator delivering parcels in the last-mile segment of the supply chain. Our working hypothesis is that the relevant decisions are affected by two main trade-offs: workforce size and shift stability. A large workforce is able to deal with demand fluctuations but incurs … Read more