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