Applying a Random-Key Optimizer on Mixed Integer Programs

Abstract

Mixed-Integer Programs (MIPs) are NP-hard optimization models that arise in a broad range of decision-making applications, including finance, logistics, energy systems, and network design. Although modern commercial solvers have achieved remarkable progress and perform effectively on many small- and medium-sized instances, their performance often degrades when confronted with large-cale or highly constrained formulations. This paper explores the use of the Random-Key Optimizer (RKO) framework as a flexible, metaheuristic alternative for computing high-quality solutions to MIPs through the design of problem-specific decoders. The proposed approach separates the search process from feasibility enforcement by operating in a continuous random-key space while mapping candidate solutions to feasible integer solutions via efficient decoding procedures. We evaluate the methodology on two representative and structurally distinct benchmark problems: the mean-variance Markowitz portfolio optimization problem with buy-in and cardinality constraints, and the Time-Dependent Traveling Salesman Problem. For each formulation, tailored decoders are developed to reduce the effective search space, promote feasibility, and accelerate convergence. Computational experiments demonstrate that RKO consistently produces competitive, and in several cases superior, solutions compared to a state-of-the-art commercial MIP solver, both in terms of solution quality and computational time. These results highlight the potential of RKO as a scalable and versatile heuristic framework for tackling challenging large-scale MIPs.

Figures (8)

• Figure 1: Schematic illustration of our approach. Our framework is based on a distinct separation of problem-independent (blue hypercube) and problem-dependent (decoder) modules. The search for high-quality solutions is performed within the (problem-independent) space of random keys, as can be implemented with a stack of algorithmic paradigms such as genetic algorithms, simulated annealing, or swarm-based algorithms. The deterministic (problem-dependent) decoder maps a given random-key vector $\mathcal{X} \in [0,1)^n$ to a feasible solution $x=\mathit{decoder(\mathcal{X})}$ within the problem space (as displayed by black dots). For mixed integer programs, our decoder design allows for an efficient, native encoding of a plethora of common constraints such as integrality constraints, variable bounds, and cardinality constraints.
• Figure 2: Decoding example. The selected assets are 9, 3, and 6.
• Figure 3: Decoding example. The selected customer visiting sequence $5-4-2-3-1-6$.
• Figure 4: Time-to-target plot for instance Russell 3000 with $K=40$ and $\lambda=0.5$.
• Figure 5: Time-to-target plot for instance Russell 3000 with $K=60$ and $\lambda=0.5$.