arXiv subjects

Hugh Medal

Publications and source records attributed to Hugh Medal.

4 recordsLinked to original sources

arXiv2024

A Successive Refinement Algorithm for Tri-Level Stochastic Defender-Attacker Problems with Decision-Dependent Probability Distributions

Tri-level defender-attacker game models are a well-studied method for determining how best to protect a system (e.g., a transportation network) from attacks. Existing models assume that defender and attacker actions have a perfect effect, i.e., system components hardened by a defender cannot be destroyed by the attacker, and attacked components always fail. Because of these assumptions, these models produce solutions in which defended components are never attacked, a result that may not be realistic in some contexts. This paper considers an imperfect defender-attacker problem in which defender decisions (e.g., hardening) and attacker decisions (e.g., interdiction) have an imperfect effect such that the probability distribution of a component's capacity depends on the amount of defense and attack resource allocated to the component. Thus, this problem is a stochastic optimization problem with decision-dependent probabilities and is challenging to solve because the deterministic equivalent formulation has many high-degree multilinear terms. To address the challenges in solving this problem, we propose a successive refinement algorithm that dynamically refines the support of the random variables as needed, leveraging the fact that a less-refined support has fewer scenarios and multilinear terms and is, therefore, easier to solve. A comparison of the successive refinement algorithm versus the deterministic equivalent formulation on a tri-level stochastic maximum flow problem indicates that the proposed method solves many more problem instances and is up to $66$ times faster. These results indicate that it is now possible to solve tri-level problems with imperfect hardening and attacks.

math.OC↗

arXiv2024

A Successive Refinement for Solving Stochastic Programs with Decision-Dependent Random Capacities

We study a class of two-stage stochastic programs in which the second stage includes a set of components with uncertain capacity, and the expression for the distribution function of the uncertain capacity includes first-stage variables. Thus, this class of problems has the characteristics of a stochastic program with decision-dependent uncertainty. A natural way to formulate this class of problems is to enumerate the scenarios and express the probability of each scenario as a product of the first-stage decision variables; unfortunately, this formulation results in an intractable model with a large number of variable products with high-degree. After identifying structural results related to upper and lower bounds and how to improve these bounds, we present a successive refinement algorithm that successively and dynamically tightens these bounds. Implementing the algorithm within a branch-and-cut method, we report the results of computational experiments that indicate that the successive refinement algorithm significantly outperforms a benchmark approach. Specifically, results show that the algorithm finds an optimal solution before the refined state space become too large.

math.OC↗

arXiv2023

DiversiTree: A New Method to Efficiently Compute Diverse Sets of Near-Optimal Solutions to Mixed-Integer Optimization Problems

While most methods for solving mixed-integer optimization problems compute a single optimal solution, a diverse set of near-optimal solutions can often lead to improved outcomes. We present a new method for finding a set of diverse solutions by emphasizing diversity within the search for near-optimal solutions. Specifically, within a branch-and-bound framework, we investigated parameterized node selection rules that explicitly consider diversity. Our results indicate that our approach significantly increases the diversity of the final solution set. When compared with two existing methods, our method runs with similar runtime as regular node selection methods and gives a diversity improvement between 12% and 190%. In contrast, popular node selection rules, such as best-first search, in some instances performed worse than state-of-the-art methods by more than 35% and gave an improvement of no more than 130%. Further, we find that our method is most effective when diversity in node selection is continuously emphasized after reaching a minimal depth in the tree and when the solution set has grown sufficiently large. Our method can be easily incorporated into integer programming solvers and has the potential to significantly increase the diversity of solution sets.

cs.DM↗

arXiv2022

A Spatial Branch-and-Bound Approach for Maximization of Non-Factorable Monotone Continuous Submodular Functions

In contrast to the many continuous global optimization methods that assume the objective function and constraints are factorable, we study how to find globally maximal solutions to problems that are not factorable, focusing on a particular class of problems. Specifically, we develop a method for non-decreasing continuous submodular functions subject to constraints. We characterize the hypograph of such functions and develop a cutting plane algorithm that finds approximate solutions and bounds using an approximation of the convex hull of the hypograph. We also test a spatial branch-and-bound approach that utilizes the approximate cutting plane algorithm to form an outer approximation and obtain upper bounds for a sub-rectangle and compare our method with a state-of-the-art commercial solver. The main result is that for some problems the property of submodularity is more useful than factorability.

math.OC↗