Conic formulation of QPCCs applied to truly sparse QPs
研究了带互补约束的非凸二次优化问题，在仅涉及约束的温和条件下建立了精确的完全正定重构，并给出了强锥对偶条件。该方法避免了分支和大常数，适用于追求可解释稀疏解的二次优化问题，如稀疏最小二乘回归。
Abstract We study (nonconvex) quadratic optimization problems with complementarity constraints, establishing an exact completely positive reformulation under—apparently new—mild conditions involving only the constraints, not the objective. Moreover, we also give the conditions for strong conic duality between the obtained completely positive problem and its dual. Our approach is based on purely continuous models which avoid any branching or use of large constants in implementation. An application to pursuing interpretable sparse solutions of quadratic optimization problems is shown to satisfy our settings, and therefore we link quadratic problems with an exact sparsity term $$\Vert {{\mathsf {x}}}\Vert _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mi>x</mml:mi> <mml:mo>‖</mml:mo> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> </mml:math> to copositive optimization. The covered problem class includes sparse least-squares regression under linear constraints, for instance. Numerical comparisons between our method and other approximations are reported from the perspective of the objective function value.