Solving Convex Quadratic Optimization with Indicators Over Structured Graphs

TL;DR

This paper introduces an exact dynamic programming algorithm for convex quadratic problems with indicator variables on structured graphs, achieving linear complexity under certain conditions and demonstrating practical effectiveness in forecasting and outlier detection.

Contribution

The paper develops a novel parametric dynamic programming approach that exploits graph structure to solve convex quadratic problems efficiently, extending to practical forecasting applications.

Findings

Algorithm scales linearly with problem size under structural conditions

Outperforms existing solvers on synthetic and real datasets

Effective joint forecasting and outlier detection framework

Abstract

This paper studies convex quadratic minimization problems in which each continuous variable is coupled with a binary indicator variable. We focus on the structured setting where the Hessian matrix of the quadratic term is positive definite and exhibits sparsity. We develop an exact parametric dynamic programming algorithm whose computational complexity depends explicitly on the treewidth of the Hessian's support graph, its volume growth, and an appropriate margin parameter. Under suitable structural conditions, the overall complexity scales linearly with the problem dimension. To demonstrate the practical impact of our approach, we introduce a novel framework for joint forecasting and outlier detection by extending exponential smoothing to time series with outliers. Computational experiments on both synthetic and real data sets show that our method significantly outperforms state-of-the-art solvers.