Kaixiang Hu | Searcharxiv

arXiv2025

New Lagrangian dual algorithms for solving the continuous nonlinear resource allocation problem

The continuous nonlinear resource allocation problem (CONRAP) has broad applications in economics, engineering, production and inventory management, and often serves as a subproblem in complex programming. Without relying on monotonicity assumptions for the objective and constraint functions, we propose two Lagrangian dual algorithms for solving two types of CONRAP. Both algorithms determine an update strategy for the Lagrange multiplier, utilizing the values of the objective and constraint functions at the current and previous iterations. This strategy accelerates the process of finding dual optimal solutions. Subsequently, leveraging the problem's convexity, the primal optimal solution is either directly identified or derived by solving a one-dimensional linear equation. We also prove that both algorithms converge to optimal solutions within a finite number of iterations. Numerical experiments on six types of practical test problems illustrate the superior computational efficiency of the proposed algorithms. For test problems with a general inequality constraint, the first algorithm achieves a CPU time reduction exceeding an order of magnitude compared to solvers such as Gurobi and CVX. For test problems with a linear equality constraint, the second algorithm consistently outperforms four existing algorithms, delivering an improvement of over two orders of magnitude in computational efficiency.

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arXiv2024

An exact column generation algorithm for load balancing in capacity sharing networks

Capacity sharing networks are typical heterogeneous communication networks widely applied in information and communications technology (ICT) field. In such networks, resources like bandwidth, spectrum, computation and storage are shared among various communication services. Meanwhile, the issue of network congestion is always a prominent challenge. To handle network congestion essentially needs to solve the load balancing of networks. In this paper, for capacity sharing networks, we formulate their load balancing problem as a maximum multi-commodity flow problem. For such a problem, always a large-scale linear programming, the column generation algorithm is a commonly used and crucial method to solve it. In each iteration, this algorithm involves solving a linear programming subproblem and determining whether to terminate or generate a new column for inclusion in the subproblem. This iterative procedure of solving and checking continues throughout the algorithm. Nevertheless, since the checking subproblem is NP-hard, its solution significantly impacts the overall efficiency of the algorithm. In this paper, we innovatively convert the checking subproblem into a single-constrained shortest path (SCSP) subproblem. By exactly solving the SCSP subproblem, we can obtain the optimal solution to the checking subproblem with same or less computing time. Experimental results demonstrate that our algorithm achieves computational efficiency comparable to heuristic algorithms while outperforming other state-of-the-art algorithms by at least an order of magnitude.

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