Simplicial branchandbound for quadratically constrained quadratic programs 253. Abstract in this paper, we give a new penalized semide. Pdf a branch and cut algorithm for nonconvex quadratically. Hidden convexity in some nonconvex quadratically constrained quadratic programming.
Computable by interior point method nesterov and nemirovskii 1994. Copositive programming, quadratically constrained quadratic programs, convex representation. In this paper, we study some bounds for nonconvex quadratically constrained quadratic programs. Such an nlp is called a quadratic programming qp problem. Linear programming probleminteger programming problem bounds. An iterative method for nonconvex quadratically constrained. A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Next, we discuss how to polynomially find the best reformulation corresponding with the tightest continuous bound within this class.
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. Under a suitable simultaneous diagonalization assumption which trivially holds for trust region type problems, we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Keywords adaptive ellipsoid based method quadratically constrained. A sequential quadratically constrained quadratic programming. Semide nite programming lecture 5 or 6327 spring 2012. Nemirovski interiorpoint polynomial algorithms in convex programming siam.
Mixed integer quadratically constrained programming. Examples of problems to which this idea has been successfully ap plied include. A nonconvex quadratically constrained quadratic program qcqp. Convex quadratic relaxations of nonconvex quadratically. 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.
On solving quadratically constrained quadratic programming. Constraint programming for strictly convex integer. Pdf to find a global optimal solution to the quadratically constrained quadratic. We first propose the class of qccr for the studied problem. New bounds for nonconvex quadratically constrained quadratic. Quadratically constrained quadratic programming qcqp constitutes a class of compu tationally hard optimization problems that have a broad spectrum of applications in wireless communications, networking, signal processing, power systems, and other areas. We propose to tighten the relaxation by using secondorder cone constraints, resulting in a convex quadratic relaxation.
Pdf convex hull presentation of a quadratically constrained. This paper presents a sequential quadratically constrained quadratic programming sqcqp method for solving smooth convex programs. A new convex relaxation for quadratically constrained quadratic. We only brie y mention global methods for solving qcqps in x1. The sqcqp method solves at each iteration a subproblem that invol. A quadratically constrained quadratic program qcqp 1. Note that the rank one requirement of qcqp2 is not a convex constraint.
Largescale quadratically constrained quadratic program via. We propose to tighten the relaxation by using second order cone. One of the existing approaches for solving nonconvex qcqp problems relaxes the rank one constraint on. Largescale quadratically constrained quadratic program. Nemirovski interiorpoint polynomial algorithms in convex programming siam studies in applied mathematics, 1994. Hidden convexity in some nonconvex quadratically constrained. We start in section 2 by describing several general convex. 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.
Journal of computational and applied mathematics 220. Anstreicher, journalmathematical programming, year2012, volume6, pages233251. In this article, we study the convex hull presentation of a quadratically constrained set. Recently, zamani has proposed a dual for linearly constrained quadratic programs, where lagrange multipliers are affine functions. First, we consider the problem of minimizing an indefinite quadratic objective subject to twosided indefinite quadratic constraints. An alternating projection approach for nonconvex quadratically constrained quadratic programmings. What is the fastest algorithm to solve an equalityconstrained convex quadratic program.
Difference of convex solution of quadratically constrained. Section 4 provides some numerical examples to compare the presented relaxation. How to solve this quadratically constrained quadratic. This framework can be applied to general qcqps for which there are no available specialized methods. 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. Nonconvex quadratically constrained quadratic programming. For cubic bounds, in addition to affine multipliers we employ quadratic functions. Reformulating mixedinteger quadratically constrained quadratic. Semide nite programming lecture 5 or 6327 spring 2012 february 7, 2012 scribe.
A new convex relaxation for quadratically constrained. On convex relaxations for quadratically constrained quadratic. On convex relaxations for quadratically constrained. Introduction we consider the following quadratically constrained quadratic programming, qcqp, min xtq 0x. Convex quadratically constrained quadratic programming qcqp. Keywords quadratically constrained quadratic programming. A nonconvex quadratically constrained quadratic program. We remark that software is now emerging for convex miqcqp see, e. Such a problem is more general than a qp or lp problem, but less general than a convex nonlinear problem.
The paper proposes a novel class of quadratically constrained convex reformulations qccr for semicontinuous quadratic programming. Tighter quadratically constrained convex reformulations for. A nonconvex quadratically constrained quadratic program qcqp ask question. Popular solver with an api for several programming languages.
A new linesearch method for quadratically constrained. New bounds for nonconvex quadratically constrained. Representing quadratically constrained quadratic programs as. Industrial engineeri g and mana,en ent technion i israel institute oj technology, hat 32000, lsroel h schmd o. Sequential quadratically constrained quadratic programming. Introduction q uadratically constrained quadratic programming qcqp is an optimization problem that minimizes a quadratic function subject to quadractic inequality and equality constraints 1. Applying the new result, we solve a kind of quadratically constrained quadratic programming problems. A superlinearly convergent sequential quadratically. A new linesearch method for quadratically constrained convex.
Jul 28, 2006 this paper presents a sequential quadratically constrained quadratic programming sqcqp method for solving smooth convex programs. On convex relaxations for quadratically constrained quadratic programming. Comparing convex relaxations for quadratically constrained quadratic programming kurt m. It is convex a convex function subject to a convex set. 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. 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. 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. June 10, 2011 abstract nonconvex quadratic constraints can be linearized to obtain relaxations in a wellunderstood manner. We explore, through examples, some constraint patterns for opf over. We incorporate penalty terms into the objective of convex relaxations in order to retrieve feasible and nearoptimal solutions for nonconvex qcqps. Quadratically constrained quadratic program wikipedia.
By using this method, we propose two types of bounds for quadratically constrained quadratic programs, quadratic and cubic bounds. Chaoxu tong convex quadraticallyconstrained quadratic programming. We provide a comparison between the proposed cubic bound and typical bounds for standard quadratic programs. Difference of convex solution of quadratically constrained optimization problems difference of convex solution of quadratically constrained optimization problems van voorhis, tim.
A new method is proposed for the linesearch procedure in logarithmic barrier function and other interior point methods for convex quadratically constrained quadratic programming problems, which includes linear and quadratic programming as special cases. Our quadratic approximation to the bilinear term is compared to the linear mccormick bounds. Moreover, the former phase can preserve the subfeasibility of the current iteration, and control the increase of the objective function. Request pdf a new convex relaxation for quadratically constrained quadratic programming a new relaxation strategy is presented in this paper to approximately solve the quadratically and. For a continuous optimization problem, we say it can be solved in. Convex quadratic relaxations of nonconvex quadratically constrained quadratic progams john e. Difference of convex programming techniques were used. 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. 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. Pdf in this paper, we study some bounds for nonconvex quadratically constrained quadratic programs. Quadratic programming is a particular type of nonlinear programming.
Nonconvex quadratic constraints can be linearized to obtain relaxations in a wellunderstood manner. Problems of this type have recently been studied by stern and wolkowicz 15 as a natural extension of the quadratic model qp. We consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex quadratic constraints. Therefore, our extension of qcr is likely to be of practical use. Quadratically constrained quadratic programming qcqp constitutes an important class of computationally hard optimization problems with a broad spectrum of applications ranging from wireless communications and networking, e. Pdf new bounds for nonconvex quadratically constrained. Revised, may 29, 2005 abstract in this paper we propose a smoothing newtontype algorithm for the problem of minimizing a convex quadratic function subject to. The sqcqp method solves at each iteration a subproblem that involves convex quadratic inequality constraints as well as a. General heuristics for nonconvex quadratically constrained. Moreover, we report comparison results of a quadratic and a cubic bound for some non convex quadratically constrained quadratic programs. The secondorder cone constraints are based on linear combinations. At each iteration, only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction.
Interestingly, handling quadratic constraints adequately turns out to be a nontrivial exercise. A quadratically constrained quadratic program qcqp is an optimization. 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. Hidden convexity in some nonconvex quadratically constrained quadratic programming article pdf available in mathematical programming 721. Consensusadmm for general quadratically constrained. Mathemalical programming 72 1996 5163 hidden convexity in some nonconvex quadratically constrained quadratic programming aharon bental a. While problems with a quadratic objective function and multiple quadratic constraints were introduced as early as 1951 in kt51, duality results. Applying the new result, we solve a kind of quadratically constrained quadratic programming problems, which generalizes many wellstudied problems. Reformulating mixedinteger quadratically constrained. The simplexbased methods used in the lpquadratic solver, the largescale lpqp solver, and the.
We present a branch and cut algorithm that yields in finite time, a globally. How to solve this quadratically constrained quadratic programming problem. The sqcqp method solves at each iteration a subproblem that involves convex quadratic inequality constraints as well as a convex quadratic objective function. Comparing convex relaxations for quadratically constrained. On convex relaxations for quadratically constrained quadratic programming kurt m. Numerical examples are given to show the applicability of the new method. Spacecraft attitude control under constrained zones via.
Anstreicher, journalmathematical programming, year2012, volume6, pages233251 kurt m. Consider the quadratically constrained quadratic programming qcqp. Pdf kkt solution and conic relaxation for solving quadratically. Tighter quadratically constrained convex reformulations. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. Pdf hidden convexity in some nonconvex quadratically. Outline of the paper the main goal of the paper is to present an overview of examples and appli cations of secondorder cone programming. Generalized trust region problem jin, fang and xing 2010. Quadratically constrained quadratic programs on acyclic. A simplicial branchandbound algorithm for solving quadratically constrained quadratic programs received. Quadratically constrained programming a problem with.
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