Global and Robust Optimization for Non-Convex Quadratic Programs

TL;DR

This study tackles non-convex QCQPs under convex uncertainty by developing RsBB, a unified framework that couples robust cutting planes with spatial branch-and-bound. The method uses McCormick envelopes for convex relaxations and a two-level procedure to assess robustness while guiding global search, demonstrated on pooling problems with box, ellipsoidal, and polyhedral uncertainties. Results show RsBB achieves robust optimality on the majority of cases (about 97%), offering meaningful robustness benefits though with higher computational cost compared to purely global reformulations. The work highlights the value of integrating robustness considerations into global optimization, and points to potential improvements via tighter relaxations and problem-specific relaxations to enhance scalability and applicability to broader non-convex problems.

Abstract

This paper presents a novel algorithm integrating global and robust optimization methods to solve continuous non-convex quadratic problems under convex uncertainty sets. The proposed Robust spatial branch-and-bound (RsBB) algorithm combines the principles of spatial branch-and-bound (sBB) with robust cutting planes. We apply the RsBB algorithm to quadratically constrained quadratic programming (QCQP) problems, utilizing McCormick envelopes to obtain convex lower bounds. The performance of the RsBB algorithm is compared with stateof-the-art methods that rely on global solvers. As computational test bed for our proposed approach we focus on pooling problems under different types and sizes of uncertainty sets. The findings of our work highlight the efficiency of the RsBB algorithm in terms of computational time and optimality convergence and provide insights to the advantages of combining robustness and optimality search.

Figures (10)

• Figure 1: Feasible regions for nominal sampled and robust toy problems
• Figure 2: Contour plot for toy problem over the nominal and sampled feasible regions. Rhombuses correspond to objective values found at denoted nodes.
• Figure 3: RsBB tree at termination for the toy problem. We denote as $z_n$ and $\tilde{z}_n$ the solutions of $P_{samp,toy}$ and $\tilde{P}_{samp,toy}$ at node $n$. We note as $infeas$ the nodes for which $\tilde{P}_{samp,toy}$ was found infeasible. Nodes with dashed lines correspond to fathomed nodes. Nodes with thin lines ($n=0,1$) are evaluated for the original sampled uncertainty set $\hat{\mathcal{U}}=\{4\}$. At node 2 a feasibility violation is detected and the sampled uncertainty set is updated accordingly to $\hat{\mathcal{U}}=\{4,6\}$. The remaining nodes with thick lines ($n=3,\cdots,14$) are evaluated for the updated sampled uncertainty set.
• Figure 4: Cumulative CPU time for solved instances via Dual reformulation, PyROS and RsBB methods for different uncertainty types
• Figure 5: Variability of tree nodes explored for set size and set type for problems solved to robust optimality within 1 hour.

Theorems & Definitions (20)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 1
• proof
• Lemma 2
• proof