General heuristics for nonconvex quadratically constrained quadratic programming jaehyun park stephen boyd march 22, 2017 abstract we introduce the suggestandimprove framework for general nonconvex quadratically constrained quadratic programs qcqps. An iterative method for nonconvex quadratically constrained quadratic programs chuangchuang sun and ran dai abstractthis paper examines the nonconvex quadratically constrained quadratic programming qcqp problems using an iterative method. Spacecraft attitude control under constrained zones via. If convex we can use readily available qcp socp solvers including cplex and gurobi. Feasible region objective 150 objective 169 objective 0 objective 50. This paper presents a novel optimization method for effectively solving nonconvex quadratically constrained quadratic programs nqcqp problem. Quadratically constrained programming a problem with linear and convex quadratic constraints, and a linear or convex quadratic objective, is called a quadratically constrained qcp problem. Interestingly, handling quadratic constraints adequately turns out to be a nontrivial exercise. Largescale quadratically constrained quadratic program via low. Semidefinite programming versus the reformulationlinearization technique for nonconvex quadratically constrained quadratic programming.
Reformulating mixedinteger quadratically constrained. Oct 27, 2012 we consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex quadratic constraints. Multiterm polyhedral relaxations for nonconvex, quadratically. Convex quadratic relaxations of nonconvex quadratically. On convex relaxations for quadratically constrained. A nonconvex qcqp is to minimize a quadratic function subject to a set of quadratic constraints, where the cost and constraints are not necessarily to be convex. The simplexbased methods used in the lpquadratic solver, the largescale lpqp solver, and the. One of the existing approaches for solving nonconvex qcqp problems relaxes the rank one constraint on. We propose to tighten the relaxation by using second order cone. Using the convex envelope of multilinear functions as our starting point, we develop a polyhedral relaxation for qcqp, along with a. Quadratically constrained quadratic programs on acyclic.
This article addresses the generation of strong polyhedral relaxations for nonconvex, quadratically constrained quadratic programs qcqps. Global solution of nonconvex quadratically constrained quadratic programs. A new convex relaxation for quadratically constrained quadratic programming duzhi wua, aiping hub, jie zhouc, songlin wua adepartment of fundamental studies, logistical engineering university, chongqing 400016, china bschool of mathematics and statistics, chongqing university of. Nonconvex quadratically constrained quadratic programming. Quadratically constrained quadratic program wikipedia. I have found something useful in matlab optimization toolbox, i. Popular solver with an api for several programming languages. Do you mean to look for a point of minimal norm in the intersection. The class of mixedinteger quadratically constrained quadratic programs qcqp consists of minimizing. Moreover, the former phase can preserve the subfeasibility of the current iteration, and control the increase of the objective function. On convex relaxations for quadratically constrained quadratic programming kurt m.
At the same time, we can use well known semidefinite programming relaxations for qcqps to obtain lower bounds on the optimal values 29. More specifically, we use rankone matrices and constraint matrices to decompose the indefinite quadratic objective into a d. General heuristics for nonconvex quadratically constrained. Matlab software for disciplined convex programming web page and.
Outline of the paper the main goal of the paper is to present an overview of examples and appli cations of secondorder cone programming. We consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex quadratic constraints. Anstreicher, journalmathematical programming, year2012, volume6, pages233251. If all the fi are convex then the qcqp may be solved by sdp. Convex quadratically constrained quadratic programming qcqp can be reduced to semidefinite programming sdp. A variety of relaxations for quadratically constrained quadratic programming qcqp can be formulated as semidefinite programs sdps.
We remark that software is now emerging for convex miqcqp see, e. A nonconvex quadratically constrained quadratic program qcqp ask question. A simplicial branchandbound algorithm for solving quadratically constrained quadratic programs received. Difference of convex programming techniques were used. Sequential quadratically constrained quadratic programming. Global solution of non convex quadratically constrained quadratic programs. We first propose the class of qccr for the studied problem. Solvers like cplex can solve problems with convex quadratic constraints as well. A nonconvex quadratically constrained quadratic program qcqp.
An alternating projection approach for nonconvex quadratically constrained quadratic programmings. Learning the kernel matrix in discriminant analysis via. A quadratically constrained quadratic program qcqp has the form minimize f0. A quadratically constrained quadratic program qcqp 1. A new convex relaxation for quadratically constrained quadratic programming duzhi wua, aiping hub, jie zhouc, songlin wua adepartment of fundamental studies, logistical engineering university, chongqing 400016, china bschool of mathematics and statistics, chongqing university of technology, chongqing 400054,china. If p1, pm are all zero, then the constraints are in fact linear and the. A commercial optimization solver for linear programming, nonlinear programming, mixed integer linear programming, convex quadratic programming, convex quadratically constrained quadratic programming, secondorder cone programming and their mixed integer counterparts. If not convex, we can try a general purpose nlp solver such as ipopt. Comparing convex relaxations for quadratically constrained.
Quadratic programming solvers quadratic programming solvers are used when the objective function has quadratic terms. Mixedinteger quadraticallyconstrained quadratic programs miqcqps, and related classes of problems such as quadraticallyconstrained quadratic programs qcqps and quadratic programs qps arise in a wide variety of engineering and scientific applications including pooling and blending in process. Sequential quadratic programming for quadratically constrained quadratic programs. At each iteration, only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction. Simplicial branchandbound for quadratically constrained quadratic programs 253. Robert fourer, jared erickson, convex quadratic programming in ampl iccopt 20 lisbon 29 july1 august 20 2 convex quadratic programming in ampl a surprising variety of optimization applications can be written in terms of convex quadratic objectives and constraints that are handled effectively by extensions to linear solvers. Could someone tell me the time complexity of a convex quadratically constrained quadratic program qcqp. You could certainly give it to a more general purpose branch and bound code for nonconvex minlp problems like baron. Quadratically constrained programming a problem with linear. All these answer seem to forget one important thing. A nonconvex qcqp is to minimize a quadratic function subject to a set of quadratic constraints, where the. A novel optimization method for nonconvex quadratically. If yes, this is a convex problem, and your formulation has a bug, it appears. Such a problem is more general than a qp or lp problem, but less general than a convex nonlinear problem.
Solving a nonconvex quadraticallyconstrained quadratic program. Using the convex envelope of multilinear functions as our starting point, we develop a polyhedral relaxation for qcqp, along with a cutting plane algorithm for its implementation. Robust quadratic programming with mixedinteger uncertainty areesh mittal, can gokalp, and grani a. An interior point method for quadratically constrained convex quadratic programming is presented that is based on a logarithmic barrier function approach and terminates at a required accuracy of an approximate solution in polynomial time. Home browse by title periodicals journal of global optimization vol. Hanasusanto graduate program in operations research and industrial engineering, the university of texas at austin, usa december 17, 2018 abstract we study robust convex quadratic programs where the uncertain problem parameters can contain. Tighter quadratically constrained convex reformulations. We introduce the suggestandimprove framework for general nonconvex quadratically constrained quadratic programs qcqps. These problems can also be formulated as secondorder cone programs socps, including formulations with rotated cones.
Enhancing relaxations for nonconvex mixedinteger quadratically constrained quadratic programs. Quadratically constrainted quadratic programming qcqp in. Global solution of nonconvex quadratically constrained. Solution of monotone complementarity and general convex. Global solution of nonconvex quadratically constrained quadratic. Therefore, our extension of qcr is likely to be of practical use. So given your functions are positive semidefinite, there are a number of algorithms you can use see. T1 solution of monotone complementarity and general convex programming problems using a modified potential reduction interior point method.
The primary purpose of this paper is to present a systematic. When the decision variables can be either continuous or integer, mixed integer quadratic programming solvers miqp are used. The paper proposes a novel class of quadratically constrained convex reformulations qccr for semicontinuous quadratic programming. On convex relaxations for quadratically constrained quadratic. For the nonconvex case, you can of course apply an sqp method, but it would not be. Next, we discuss how to polynomially find the best reformulation corresponding with the tightest continuous bound within this class. Quadratic programming qp is the process of solving a special type of mathematical optimization problemspecifically, a linearly constrained quadratic optimization problem, that is, the problem of optimizing minimizing or maximizing a quadratic function of several variables subject to linear constraints on these variables.
Comparing convex relaxations for quadratically constrained quadratic programming kurt m. Quadratically constrainted quadratic programming qcqp in matlab. June 10, 2011 abstract nonconvex quadratic constraints can be linearized to obtain relaxations in a wellunderstood manner. Low, fellow, ieee abstractwe prove that nonconvex quadratically constrained quadratic programs can be solved in polynomial time when. An iterative method for nonconvex quadratically constrained. If each of these matrices are positive definite, then the optimization problem is convex. N2 we present a homogeneous algorithm equipped with a modified potential function for the monotone complementarity problem. A quadratically constrained quadratic program qcqp is an optimization.
On computing the center of a convex quadratically constrained set. Using this framework, we generalize a number of known methods and provide heuristics to get approximate solutions to qcqps for which no specialized methods are available. Convex relaxations for nonconvex quadratically constrained quadratic. Anstreicher department of management sciences university of iowa iowa city, ia 52242 usa july 28, 2010 abstract we consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex. Sequential quadratic programming for quadratically. Quadratically constrained quadratic programming qcqp. Complexity of convex quadratically constrained quadratic. If these matrices are neither positive nor negative semidefinite, the problem is nonconvex. Consider a quadratically constrained quadratic program. Solving a nonconvex quadraticallyconstrained quadratic. They can also be used to provide approximations to non. When the decision variables can be either continuous or integer, mixed integer quadratic constrained programming solvers are used. By applying a novel parametric linearizing approach, the initial nqcqp problem and its subproblems can be transformed into a sequence of parametric linear programs relaxation problems. A logarithmic barrier function algorithm for quadratically.
Quadratically constrained quadratic programming in r. To enhance the computational efficiency of the presented algorithm. N2 we propose a method for finding analytic center of a convex feasible region whose boundaries are defined by quadratic functions. Convex quadratic relaxations of nonconvex quadratically constrained quadratic progams john e. T1 on computing the center of a convex quadratically constrained set. Robust quadratic programming with mixedinteger uncertainty. Quadratic boolean programming 1 is a particular case of quadratically constrained quadratic problems qcqp, so general heuristics for this class of problems may be applied. Proceedings of the thirteenth acm sigkdd international conference on knowledge discovery and data mining. A number of alternative methods have been proposed for strictly convex quadratic programming with a modest number of constraints and variables, see, e. Quadratically constrained quadratic programs on acyclic graphs with application to power. What is the fastest algorithm to solve an equalityconstrained convex quadratic program. How to solve this quadratically constrained quadratic.
We start in section 2 by describing several general convex. In general, however, solving qcqp is nphard, which can be verified by. Methods for convex and general quadratic programming. Recently i have run into a quadratically constrainted quadratic programming qcqp problem in my research. Convex optimization mlss 2009 convex sets and functions. Quadratically constrained programming a problem with. A quadratically constrained quadratic program qcqp is an. Tighter quadratically constrained convex reformulations for.
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