Convexification of a Separable Function over a Polyhedral Ground Set

TL;DR

This work investigates the convexification of the nonconvex set $\mathcal{S}^\kappa$ defined over a polyhedral ground set, focusing on the standard simplex. It introduces a hierarchy of conic relaxations based on power cones, second-order cones, and semidefinite programming, augmented by Reformulation-Linearization Technique (RLT) and Reformulation-Perspectification Technique (RPT), to obtain the convex hull or tight outer-approximations. For $\kappa=2$, it provides exact hulls in low dimensions and establishes approximation guarantees for general $\kappa$, including probabilistic tightness as $n\to\infty$ under uniform objectives. Computational experiments compare nine relaxations, showing that power-cone relaxations are strong and that RLT and RPT-based relaxations offer substantial improvements, with exactness achieved in some small-dimensional cases. The results advance tractable convexification methods for separable nonconvex functions over polyhedral ground sets and offer practical guidance for optimization in related network-flow contexts.

Abstract

In this paper, we study the set $\mathcal{S}^κ= \{ (x,y)\in\mathcal{G}\times\mathbb{R}^n : y_j = x_j^κ, j=1,\dots,n\}$, where $κ> 1$ and the ground set $\mathcal{G}$ is a nonempty polytope contained in $[0,1]^n$. This nonconvex set is closely related to separable standard quadratic programming and appears as a substructure in potential-based network flow problems from gas and water networks. Our aim is to obtain the convex hull of $\mathcal{S}^κ$ or its tight outer-approximation for the special case when the ground set $\mathcal{G}$ is the standard simplex. We propose power cone, second-order cone and semidefinite programming relaxations for this purpose, which are further strengthened by the Reformulation-Linearization Technique and the Reformulation-Perspectification Technique. For $κ=2$, we obtain the convex hull of $\mathcal{S}^κ$ in the low-dimensional setting. For general $κ$, we give approximation guarantees for the power cone representable relaxation, the weakest relaxation we consider. We prove that this weakest relaxation is tight with probability one as $n\to\infty$ when a uniformly generated linear objective is optimized over it. Finally, we provide the results of our extensive computational experiments comparing the empirical strength of several conic programming relaxations that we propose.

Figures (6)

• Figure 1: An example node-based structure with $n=4$ adjacent edges to node 0.
• Figure 2: Bounds derived in Propositions \ref{['prop:dbound-stSimplex-ub']} and \ref{['prop:dbound-stSimplex-lb']} for different $\kappa$ and $n$ values.
• Figure 3: Average Cumulative Gap vs. Average Cumulative Time with respect to all settings considered. The average number of exact instances is given in parenthesis.
• Figure 4: Average Cumulative Gap vs. Average Cumulative Time with respect to different distributions for the objective function coefficients.
• Figure 5: Average Cumulative Gap vs. Average Cumulative Time with respect to different $\kappa$ values.

Theorems & Definitions (48)

• Proposition 1
• proof
• Definition 1: Power cone
• Lemma 1
• Proposition 2
• proof
• Corollary 1
• Proposition 3
• Remark 1
• Proposition 4