Convexification of multi-period quadratic programs with indicators
研究了状态随线性动态演化的多期混合整数凸二次优化问题，通过投影状态变量构造闭凸包，并推导出紧的二阶锥规划公式和多项式时间算法，适用于统计学习与混合系统控制。
Abstract We study a multi-period mixed-integer convex quadratic optimization problem, where the state evolves dynamically as an affine function of the state and action (control) variables in each period. We begin by projecting out the state variables using linear dynamics, resulting in a mixed-integer quadratic optimization problem with a positive-definite (block-)factorizable cost matrix. Employing this expression, we construct a closed convex hull representation of the epigraph of the quadratic cost over the feasible region in an extended space. Subsequently, we establish a tight second-order cone programming formulation with $$\mathscr {O}(n^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> conic constraints. We further propose a polynomial-time algorithm based on a reformulation of the problem as a shortest path problem on a directed acyclic graph. To illustrate the applicability of our results across diverse domains, we present case studies in statistical learning and hybrid system control.