arXiv subjects

Minh Hoang Ha

Publications and source records attributed to Minh Hoang Ha.

6 recordsLinked to original sources

arXiv2026

Joint Binary-Continuous Fractional Programming: Solution Methods and Applications

In this paper, we investigate a class of non-convex sum-of-ratios programs relevant to decision-making in key areas such as product assortment and pricing, and facility location and cost planning. These optimization problems, characterized by both continuous and binary decision variables, are highly non-convex and challenging to solve. To the best of our knowledge, no existing methods can efficiently solve these problems to near-optimality with arbitrary precision. To address this challenge, we propose an innovative approach based on logarithmic transformations and piecewise linear approximation (PWLA) to approximate the nonlinear fractional program as a mixed-integer convex program with arbitrary precision, which can be efficiently solved using cutting plane (CP) or Branch-and-Cut (B&C) procedures. Our method offers several advantages: it allows for a shared set of binary variables to approximate nonlinear terms and employs an optimal set of breakpoints to approximate other non-convex terms in the reformulation, resulting in an approximate model that is minimal in size. Furthermore, we provide a theoretical analysis of the approximation errors associated with the solutions derived from the approximated problem. We demonstrate the applicability of our approach to constrained competitive joint facility location and cost optimization, as well as constrained product assortment and pricing problems. Extensive experiments on instances of varying sizes, comparing our method with several alternatives, including general-purpose solvers and more direct PWLA-based approximations, show that our approach consistently achieves superior performance across all baselines, particularly in large-scale instances.

math.OC↗

arXiv2025

Equilibrium-Constrained Estimation of Recursive Logit Choice Models

The recursive logit (RL) model provides a flexible framework for modeling sequential decision-making in transportation and choice networks, with important applications in route choice analysis, multiple discrete choice problems, and activity-based travel demand modeling. Despite its versatility, estimation of the RL model typically relies on nested fixed-point (NFXP) algorithms that are computationally expensive and prone to numerical instability. We propose a new approach that reformulates the maximum likelihood estimation problem as an optimization problem with equilibrium constraints, where both the structural parameters and the value functions are treated as decision variables. We further show that this formulation can be equivalently transformed into a conic optimization problem with exponential cones, enabling efficient solution using modern conic solvers such as MOSEK. Experiments on synthetic and real-world datasets demonstrate that our convex reformulation achieves accuracy comparable to traditional methods while offering significant improvements in computational stability and efficiency, thereby providing a practical and scalable alternative for recursive logit model estimation.

econ.EM↗

arXiv2024

Competitive Facility Location with Market Expansion and Customer-centric Objective

We study a competitive facility location problem, where customer behavior is modeled and predicted using a discrete choice random utility model. The goal is to strategically place new facilities to maximize the overall captured customer demand in a competitive marketplace. In this work, we introduce two novel considerations. First, the total customer demand in the market is not fixed but is modeled as an increasing function of the customers' total utilities. Second, we incorporate a new term into the objective function, aiming to balance the firm's benefits and customer satisfaction. Our new formulation exhibits a highly nonlinear structure and is not directly solved by existing approaches. To address this, we first demonstrate that, under a concave market expansion function, the objective function is concave and submodular, allowing for a $(1-1/e)$ approximation solution by a simple polynomial-time greedy algorithm. We then develop a new method, called Inner-approximation, which enables us to approximate the mixed-integer nonlinear problem (MINLP), with arbitrary precision, by an MILP without introducing additional integer variables. We further demonstrate that our inner-approximation method consistently yields lower approximations than the outer-approximation methods typically used in the literature. Moreover, we extend our settings by considering a\textit{ general (non-concave)} market-expansion function and show that the Inner-approximation mechanism enables us to approximate the resulting MINLP, with arbitrary precision, by an MILP. To further enhance this MILP, we show how to significantly reduce the number of additional binary variables by leveraging concave areas of the objective function. Extensive experiments demonstrate the efficiency of our approaches.

math.OC↗

arXiv2024

Competitive Facility Location under Cross-Nested Logit Customer Choice Model: Hardness and Exact Approaches

We study the competitive facility location problem, where a firm aims to establish new facilities in a market already occupied by competitors. In this problem, customer behavior is crucial for making optimal location decisions. We explore a general class of customer choice models, known as the cross-nested logit (CNL) model, which is recognized for its flexibility and generality in predicting people's choice behavior. To explore the problem, we first demonstrate that it is NP-hard, even when there is only one customer class. We further show that this hardness result is tight, as the facility location problem under any simpler choice models (such as the logit or nested logit) is polynomial-time solvable when there is one customer class. To tackle the resulting facility location problem, we demonstrate that the objective function under a general cross-nested structure is not concave. Interestingly, we show that by a change of variables, the objective function can be converted to a convex program (i.e., a maximization problem with a concave objective and convex constraints), enabling it to be solved to optimality via an outer-approximation algorithm. Extensive experiments show the efficiency of our approach and provide analyses on the benefits of using the cross-nested model in the facility location context.

math.OC↗

arXiv2024

Competitive Facility Location under Random Utilities and Routing Constraints

In this paper, we study a facility location problem within a competitive market context, where customer demand is predicted by a random utility choice model. Unlike prior research, which primarily focuses on simple constraints such as a cardinality constraint on the number of selected locations, we introduce routing constraints that necessitate the selection of locations in a manner that guarantees the existence of a tour visiting all chosen locations while adhering to a specified tour length upper bound. Such routing constraints find crucial applications in various real-world scenarios. The problem at hand features a non-linear objective function, resulting from the utilization of random utilities, together with complex routing constraints, making it computationally challenging. To tackle this problem, we explore three types of valid cuts, namely, outer-approximation and submodular cuts to handle the nonlinear objective function, as well as sub-tour elimination cuts to address the complex routing constraints. These lead to the development of two exact solution methods: a nested cutting plane and nested branch-and-cut algorithms, where these valid cuts are iteratively added to a master problem through two nested loops. We also prove that our nested cutting plane method always converges to optimality after a finite number of iterations. Furthermore, we develop a local search-based metaheuristic tailored for solving large-scale instances and show its pros and cons compared to exact methods. Extensive experiments are conducted on problem instances of varying sizes, demonstrating that our approach excels in terms of solution quality and computation time when compared to other baseline approaches.

cs.AI↗

arXiv2023

Optimal Privacy Preserving for Federated Learning in Mobile Edge Computing

Federated Learning (FL) with quantization and deliberately added noise over wireless networks is a promising approach to preserve user differential privacy (DP) while reducing wireless resources. Specifically, an FL process can be fused with quantized Binomial mechanism-based updates contributed by multiple users. However, optimizing quantization parameters, communication resources (e.g., transmit power, bandwidth, and quantization bits), and the added noise to guarantee the DP requirement and performance of the learned FL model remains an open and challenging problem. This article aims to jointly optimize the quantization and Binomial mechanism parameters and communication resources to maximize the convergence rate under the constraints of the wireless network and DP requirement. To that end, we first derive a novel DP budget estimation of the FL with quantization/noise that is tighter than the state-of-the-art bound. We then provide a theoretical bound on the convergence rate. This theoretical bound is decomposed into two components, including the variance of the global gradient and the quadratic bias that can be minimized by optimizing the communication resources, and quantization/noise parameters. The resulting optimization turns out to be a Mixed-Integer Non-linear Programming (MINLP) problem. To tackle it, we first transform this MINLP problem into a new problem whose solutions are proved to be the optimal solutions of the original one. We then propose an approximate algorithm to solve the transformed problem with an arbitrary relative error guarantee. Extensive simulations show that under the same wireless resource constraints and DP protection requirements, the proposed approximate algorithm achieves an accuracy close to the accuracy of the conventional FL without quantization/noise. The results can achieve a higher convergence rate while preserving users' privacy.

cs.LG↗