Dynamic programming and dimensionality in convex stochastic control
ArticleDownload View PDF
A Decomposition Framework for Nonlinear Nonconvex Two-Stage Optimization
We propose a new decomposition framework for continuous nonlinear constrained two-stage optimization, where both first- and second-stage problems can be nonconvex. A smoothing technique based on an interior-point formulation renders the optimal solution of the second-stage problem differentiable with respect to the first-stage parameters. As a consequence, efficient off-the-shelf optimization packages can be utilized. We … Read more
Fully Adaptive Zeroth-Order Method for Minimizing Functions with Compressible Gradients
We propose an adaptive zeroth-order method for minimizing differentiable functions with L-Lipschitz continuous gradients. The method is designed to take advantage of the eventual compressibility of the gradient of the objective function, but it does not require knowledge of the approximate sparsity level s or the Lipschitz constant L of the gradient. We show that … Read more
Variable metric proximal stochastic gradient methods with additional sampling
Regularized empirical risk minimization problems arise in a variety of applications, including machine learning, signal processing, and image processing. Proximal stochastic gradient algorithms are a standard approach to solve these problems due to their low computational cost per iteration and a relatively simple implementation. This paper introduces a class of proximal stochastic gradient methods built … Read more
Descent Scheme for a Class of Bilevel Programming Problems
In this paper, a class of bilevel programming problems is studied, in which the lower level is a quadratic programming problem, and the upper level problem consists of a nonlinear objective function with coupling constraints. An iterative process is developed to generate a sequence of points, which converges to the solution of this problem. In … Read more
Strengthening Dual Bounds for Multicommodity Capacitated Network Design with Unsplittable Flow Constraints
Multicommodity capacitated network design (MCND) models can be used to optimize the consolidation of shipments within e-commerce fulfillment networks. In practice, fulfillment networks require that shipments with the same origin and destination follow the same transfer path. This unsplittable flow requirement complicates the MCND problem, requiring integer programming (IP) formulations with binary variables replacing continuous … Read more
Lagrangian Duality for Mixed-Integer Semidefinite Programming: Theory and Algorithms
This paper presents the Lagrangian duality theory for mixed-integer semidefinite programming (MISDP). We derive the Lagrangian dual problem and prove that the resulting Lagrangian dual bound dominates the bound obtained from the continuous relaxation of the MISDP problem. We present a hierarchy of Lagrangian dual bounds by exploiting the theory of integer positive semidefinite matrices … Read more
Risk-Averse Antibiotics Time Machine Problem
Antibiotic resistance, which is a serious healthcare issue, emerges due to uncontrolled and repeated antibiotic use that causes bacteria to mutate and develop resistance to antibiotics. The Antibiotics Time Machine Problem aims to come up with treatment plans that maximize the probability of reversing these mutations. Motivated by the severity of the problem, we develop … Read more
A stochastic Lagrangian-based method for nonconvex optimization with nonlinear constraints
The Augmented Lagrangian Method (ALM) is one of the most common approaches for solving linear and nonlinear constrained problems. However, for non-convex objectives, handling non-linear inequality constraints remains challenging. In this paper, we propose a stochastic ALM with Backtracking Line Search that performs on a subset (mini-batch) of randomly selected points for the solving of … Read more
New Dynamic Discretization Discovery Strategies for Continuous-Time Service Network Design
Service Network Design Problems (SNDPs) are prevalent in the freight industry. While the classic SNDP is defined on a discretized planning horizon with integral time units, the Continuous-Time SNDP (CTSNDP) uses a continuous-time horizon to avoid discretization errors. Existing CTSNDP algorithms primarily rely on the Dynamic Discretization Discovery (DDD) framework, which iteratively refines discretization and … Read more