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

TL;DR

This paper introduces two novel Lagrangian dual algorithms for solving the continuous nonlinear resource allocation problem without monotonicity assumptions, demonstrating significant computational efficiency improvements over existing methods.

Contribution

The paper proposes two new Lagrangian dual algorithms that accelerate convergence and improve computational efficiency for solving CONRAP without monotonicity assumptions.

Findings

Algorithms converge to optimal solutions in finite iterations.

Significant CPU time reductions over Gurobi and CVX.

Outperform existing algorithms by over two orders of magnitude.

Abstract

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.