Cone Relaxation Research Articles

KKT Solution and Conic Relaxation for Solving Quadratically Constrained Quadratic Programming Problems

To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature. Moreover, we propose a computational scheme that provides clues of designing effective algorithms for more solvable quadratically constrained quadratic programming problems.

Full transient response of Taylor cones to a step change in electric field

We studied experimentally the complete transient response of Taylor cones subject to a step change in external electric field with the goal of finding optimal conditions to reduce the overall response time and achieve the highest possible switching bandwidth. The transient behavior of electrified menisci is of interest for many applications that would benefit from active control of on/off switching of the electrospray, such as femtoliter droplet-on-demand or novel fuel injectors in next generation internal combustion engines. We first investigated the transient behavior of ethanol, a typical solvent for droplet-on-demand. We then expanded the study to fuels such as JP-8 and E-30 biogas, a biofuel with 30% ethanol (vol.). The system response is a multi-stage process that can last from ~100 μs to ~100 ms. Potential bottleneck stages include liquid accumulation, meniscus oscillation, and cone relaxation, depending on the experimental conditions. A typical full response time is ~1 ms, and the shortest transient process observed is ~400 μs. For a given liquid, nozzle outer diameter (OD) and applied voltage are the two most important parameters to influence the full response time. Onset or near-onset voltage for the establishment of the cone jet often leads to a large number of oscillation cycles and should be avoided. Changes in conductivity and viscosity by less than a factor of 10 have negligible effects on the transient process. Using JP-8 or E-30 biogas, 90 μm OD nozzle with extractor, and flow rate of 0.4 mL/h, we can routinely achieve bandwidth of 1 kHz, corresponding to a full response time of 1 ms, after which quasi-monodispersed droplets of ~10 μm are generated. Adaptation of an inviscid model of a charged oscillating droplet to the oscillating meniscus satisfactorily explains several key phenomena observed in our experiments, such as the full response time and the overshoot of the meniscus height.

Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.

Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs

Several types of relaxations for binary quadratic polynomial programs can be obtained using linear, second-order cone, or semidefinite techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our framework, we re-derive previous relaxation schemes and provide new ones. In particular, we present three relaxations for binary quadratic polynomial programs. The first two relaxations, based on second-order cone and semidefinite programming, represent a significant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems. From a practical point of view, due to the computational cost, semidefinite-based relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances. To improve the computational efficiency for solving such problems, we propose a third relaxation based purely on second-order cone programming. Computational tests on different classes of nonconvex binary quadratic polynomial problems, including quadratic knapsack problems, show that the second-order-cone-based relaxation outperforms the semidefinite-based relaxations that are proposed in the literature in terms of computational efficiency, and it is comparable in terms of bounds.

Copositive and semidefinite relaxations of the quadratic assignment problem

Semidefinite relaxations of the quadratic assignment problem ( Q A P ) have recently turned out to provide good approximations to the optimal value of Q A P . We take a systematic look at various conic relaxations of Q A P . We first show that Q A P can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the well-studied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size.