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

TL;DR

The paper tackles the continuous nonlinear resource allocation problem (CONRAP) with convex, differentiable, separable components under general inequality and linear equality constraints. It introduces two Lagrangian dual algorithms that do not rely on monotonicity, leveraging a multiplier update strategy based on current and previous iterates and decomposing the dual into $n$ one-dimensional subproblems to enable fast solutions. The authors prove finite termination and optimality (via the KKT conditions) for both problem types and demonstrate substantial computational speedups over Gurobi, CVX, and pegging/Aug_lag across six practical problem families, including very large-scale cases. This work broadens the applicability of Lagrangian-dual methods to nonmonotone convex CONRAP and offers practical benefits for applications in economics, production, and statistics.

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.

Figures (5)

• Figure 1: In this figure, the vertical coordinate axis consists of the function $\phi(\boldsymbol{x})$ value while the horizontal coordinate axis consists of the function $g(\boldsymbol{x})$ value, the initial iterates $\boldsymbol{x_{g}} = \underset{\boldsymbol{x} \in \boldsymbol{X}}{\operatorname{argmin}\ }g(\boldsymbol{x})$ and $\boldsymbol{x_{\phi}} = \underset{\boldsymbol{x} \in \boldsymbol{X}}{\operatorname{argmin}\ }\phi(\boldsymbol{x})$, and each iterate $\boldsymbol{x}$ fulfills the box constraints $\boldsymbol{x} \in \boldsymbol{X}$.
• Figure 2: In this figure, the vertical coordinate axis consists of the function $\phi(\boldsymbol{x})$ value while the horizontal coordinate axis consists of the function $g(\boldsymbol{x})$ value, and each iterate $x$ fulfills the box constraints, $\boldsymbol{x} \in \boldsymbol{X}$. And $\boldsymbol{x_\phi} = \underset{\boldsymbol{x} \in \boldsymbol{X}}{\operatorname{argmin}\ }\phi(\boldsymbol{x})$.
• Figure 3: Perfomance profile of CPU time for $30$ random commodity warehousing test problems with a problem size of $n=2\times 10^3$.
• Figure 4: Perfomance profile of CPU time for $30$ random portfolio investment test problems with a problem size of $n=2\times 10^4$.
• Figure 5: Perfomance profile of CPU time for $30$ random sampling test problems with a problem size of $n=2\times 10^4$.

Theorems & Definitions (9)

• Lemma 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Theorem 6
• Corollary 1
• Lemma 7
• Corollary 2