One of the earliest significant results in this area is the celebrated slemma of yakubovich which plays a major role in the development of quadratic optimization. This paper introduces a new global optimization algorithm for this problem, which combines two ideas from the literaturenite branching. Nonconvex quadratic programming with box constraints is very hard to solve to proven optimality, in both theory and practice. New bounds for nonconvex quadratically constrained quadratic programming moslem zamaniabc a parametric multiobjective optimization research group, ton duc thang university, ho chi minh city, vietnam. Pdf globally solving nonconvex quadratic programs via. A number of alternative methods have been proposed for strictly convex quadratic programming with a modest number of constraints and variables, see, e. Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Cplex the ability to solve both convex and nonconvex mixedinteger quadratic programming miqp problems to proven optimality goes back few years, but still presents unclear aspects. Quadratic programming is a particular type of nonlinear programming. Be aware that also gurobi will soon have support for binary non convex quadratic. A computational study 3 and a cardinality constraint.
An accelerating algorithm for globally solving nonconvex quadratic. Solving standard quadratic programming by cutting planes. Since the authors are involved with the development of cplex, i guess this paper is relevant to your question. We are interested in understanding whether for solving an miqp problem it is favorable to linearize its quadratic part or not. A sequential quadratic programming algorithm for nonconvex. Industrial engineeri g and mana,en ent technion i israel institute oj technology, hat 32000, lsroel h schmd o. A decomposition method for nonconvex quadratically. Cq whose objective and constraints are linear in x. If the problem contains an objective function with no quadratic term, a linear objective, then the problem is termed a. Global optimization, kkt, kktqp, moment relaxations, nonconvex quadratic programming, semidefinite programming. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. 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.
Kkt, kktqp, moment relaxations, nonconvex quadratic programming, semidefinite programming. The following topics address the question of how to distinguish such problems and describe the facilities. Nonconvex quadratic programming with box constraints is a fundamental problem in the global optimisation literature, being one of the simplest n phard nonlinear programs. Nonconvex quadratic programs qps and mixedinteger quadratic programs miqps arise in a wide variety of scientific and engineering applications including facility location 21, quadratic assignment 9, molecular conformation and maxcut problems 8. Quadratic programming qp is a wellstudied fundamental nphard optimization problem which optimizes a quadratic objective over. Nonconvex qcqp represents a broad class of optimization problems. Nonconvex quadratic programming is nphardeven for the calculation of a local minimizer 44,73. Mathematical programs are a class of optimization problems with a goal of maximizing or minimizing an objective function with respect to a set of decision variables, subject to constraints on those decision variables. Globally solving nonconvex quadratic programming problems via completely positive programming jieqiu cheny samuel burerz august 15, 2011 abstract nonconvex quadratic programming qp is an nphard problem that optimizes a general quadratic function over linear constraints. A nonconvex quadratically constrained quadratic program. It is designed to solve efficiently convex problems but cannot be applied for nonconvex problems because the quadratic approximation is not positive semidefinite for nonconvex functions. More specifically, we use rankone matrices and constraint matrices to decompose the indefinite quadratic objective into a d. Multiterm polyhedral relaxations for nonconvex, quadratically. The functions defining the problem can be nonlinear and nonconvex, but must be differentiable.
We also introduce an opensource python package qcqp, which implements the heuristics discussed. 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. Globally solving nonconvex quadratic programs via linear integer programming techniques article pdf available in informs journal on computing november. Even though we set up 1 in terms of inequality constraints only, it also allows quadratic equality constraints of the form h i x 0 to be added, as they can be expressed as two quadratic inequality constraints. We have presented a heuristic that is easy to understand and fairly easy to implement using readily available software.
Globally solving nonconvex quadratic programs via linear integer. New bounds for nonconvex quadratically constrained. This article addresses the generation of strong polyhedral relaxations for nonconvex, quadratically constrained quadratic programs qcqps. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Nonconvex optimization, quadratic programming, mixed integer linear programming, global optimization. A common question i get is along the lines how can i solve a nonconvex qp using sedumi. This paper focuses on the contributions to nonconvex quadratic program.
On the copositive representation of binary and continuous. A nonconvex quadratically constrained quadratic program qcqp. On the solution of nonconvex cardinality boolean quadratic programming problems ricardo m. Second order cone programming relaxation of nonconvex. Globally solving nonconvex quadratic programming problems. On the solution of nonconvex cardinality boolean quadratic programming problems. Quadratic programming qp is a wellstudied fundamental nphard optimization problem which optimizes a quadratic objective over a set of linear constraints. Cvx, a matlab toolbox for disciplined convex programming. Global solution of nonconvex quadratically constrained. This paper presents a novel range division and contraction approach for globally solving nonconvex quadratic program with quadratic constraints. Associated with every copositive program is a dual.
Burer 3 shows that such problems can be formulated as cp programs, which are convex programs that optimize a linear function over the convex cone of completely positive matrices subject to linear constraints. The sqplabpronounce sqplab software presented in these pages is a modest matlab implementation of the sqp algorithm for solving constrained optimization problems. Quadratic programming and affine variational inequalities. Globally solving nonconvex quadratic programming problems 35 complementarity constraints. Second order cone programming relaxation of nonconvex quadratic optimization problems article pdf available in optimization methods and software 1534 august 2000 with 127 reads. Semidefinite relaxations and thus sedumi can be used to compute lower bounds on the achievable objective.
Quadratic optimization comprises one of the most important areas of nonlinear programming. Using the convex envelope of multilinear functions as our starting point, we develop a polyhedral relaxation for qcqp, along with a. We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. We are going to be dealing with quadratic programs. Many nonlinear optimization solvers search for locally optimal solutions to these problems. A linear copositive program cop is any optimization over x.
Second order cone programming relaxation of nonconvex quadratic optimization problems. By constructing new underestimating linear relaxation functions, we can transform the initial nonconvex quadratic program problem into a linear program relaxation problem. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. Global solution of nonconvex quadratically constrained quadratic programs. 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. One of the existing approaches for solving nonconvex qcqp problems relaxes the rank one constraint on. 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. Box constrained quadratic programming with proportioning. Accelerated gradient methods for nonconvex nonlinear and. Nonconvex quadratically constrained quadratic programming. Keywords nonconvex quadratic programming global optimization branchandbound semidefinite programming copositive programming completely positive programming mathematics subject classi.
On affine scaling algorithms for nonconvex quadratic. Aug 31, 2011 nonconvex quadratic programming comparisons tags. A standard quadratic program is an optimization problem that consists of minimizing a nonconvex quadratic form over the unit simplex. In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddlepoint global optimality conditions and a generalized lagrangian duality theorem for not necessarily convex polynomial optimization problems, where the lagrange multipliers are polynomials. General heuristics for nonconvex quadratically constrained. 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. Solving boxconstrained nonconvex quadratic programs pierre bonami oktay gunl uk je linderoth june, 2016 abstract we present e ective computational techniques for solving nonconvex quadratic programs with box constraints boxqp. Many classes of convex optimization problems admit polynomialtime algorithms, whereas mathematical optimization is in general nphard. This problem or variants with small differences are known under different names. Nonconvex quadratic programming comparisons yalmip. A range division and contraction approach for nonconvex. Exploiting structure in nonconvex quadratic optimization opus 4. Global optimization, kkt, kktqp, moment relaxations, nonconvex quadratic programming, semidefinite programming updated.
Quadratic objective term, specified as a symmetric real matrix. Solving nonconvex nonlinear programs with reverse convex. To globally solve a nonconvex quadratic programming problem, this. Describes solving quadratic programming problems qps with cplex. 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. As introduced in the topic stating a mip problem, a mixed integer programming mip problem can contain both integer and continuous variables. We rst observe that cutting planes obtained from the boolean quadric polytope bqp. Quadratic programs and affine variational inequalities represent two fundamental, closelyrelated classes of problems in the t,heories of mathematical programming and variational inequalities, resp tively. It is well known that a qcqp can be transformed into a rankone constrained optimization problem. On the copositive representation of binary and continuous nonconvex quadratic programs 481 note that the decomposition of nonzero x. Nonconvex polynomial programs generalized lagrangian duality global optimality sum of squares polynomials quadratic programs separable programs t. Casting the ls in problem as a nonconvex quadratically constrained quadratic program qcqp, the feasible point pursuit fpp method investigated in wang et al. The formulation p1 is known as the cardinality bqp cbqp problem.
A nonconvex quadratically constrained quadratic program qcqp ask question asked 2 years. Globally solving nonconvex quadratic programming problems via. In this paper, we reformulate qps as a mixedinteger linear problem milp. This is done via the reformulation of qp as a linear complementary problem, and the use of binary variables and bigm constraints, to model the. Nonconvex qps, however, are known to be quite hard. Generalized lagrangian duality for nonconvex polynomial. 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. A method for nonconvex quadratic programming by gennadij bulanov for windows users circut for finding approximate solutions to certain binary quadratic programs, including the maxcut and the maxbisection problems, by yin zhang the subroutines e04ncf, e04nff, e04nkf, h02cbf and h02cef from the nag fortran library.
Solving boxconstrained nonconvex quadratic programs. Distinguishes types of mixed integer programs according to quadratic terms in the objective function or constraints of the model. Box constrained quadratic programming with proportioning and. Sometimes the semidefinite relaxation is tight and a solution can be recovered.
If q is negative semidefinite, then qp is solvable. Oct 18, 2016 nonconvex quadratic programming deals with optimization problems described by means of linear and quadratic functions, i. Be aware that also gurobi will soon have support for binary non convex quadratic problem. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and. Convex programming is a subclass of nonlinear programming nlp that unifies and generalizes least squares ls, linear programming lp, and convex quadratic programming qp. We propose two alternative mixed integer linear programming formulations. August 31, 2011 a common question i get is along the lines how can i solve a nonconvex qp using sedumi the answer to the questions is a bit tricky, since it depends on what the user means with solve, and why. Mathemalical programming 72 1996 5163 hidden convexity in some nonconvex quadratically constrained quadratic programming aharon bental a. Nonconvex quadratic programming qp is an nphard problem that optimizes a general quadratic function over linear constraints. Contribute to sburerquadprogbb development by creating an account on github. General qp special case fixed cost variables outline 1 general qp introduction. This book develops a unified theory on qualitative aspects of nonconvex quadratic programming and affine variational inequ ities. It has become quite popular recently for a number of reasons, including its attractive theoretical properties, efficient numerical algorithms, and practical applications.
Globally solving nonconvex quadratic programming problems 37 2. When such problems are convex, cplex normally solves them efficiently in polynomial time. An iteration is proportional if the norm of violation of the kuhntucker conditions at active variables does not excessively exceed the norm of the part of the gradient that corresponds to free variables, while a progressive direction determines a descent direction that enables the released. Pdf second order cone programming relaxation of nonconvex. Sequential linear or quadratic programming with gradient sampling. A prototype code for nonconvex, nonsmooth constrained optimization. Nonconvex qps are nphard, and thus intractable practically impossible to solve in the general nontrivially sized case.
Accelerated gradient methods for nonconvex nonlinear and stochastic programming saeed ghadimi guanghui lan the date of receipt and acceptance should be inserted later abstract in this paper, we generalize the wellknown nesterovs accelerated gradient ag method, originally designed. The search direction computation is performed by minimizing a local linear or quadratic model of the objective subject to a linearization of the constraints. A particular attention will be paid to problems with an optimal control structure sqp stands for sequential quadratic programming. The archetypical example of such a program is the portfolio optimisation problem, as originally proposed by markowitz, in a meanvariance framework. A quadratically constrained quadratic program qcqp is an optimization. The quadratic programming solver q 2 rnn is the quadratic also known as hessian matrix a 2 rmn is the constraints matrix x 2 rn is the vector of decision variables c 2 rn is the vector of linear objective function coef. On the solution of nonconvex cardinality boolean quadratic. We connect qp with cp programs by reformulating qp as a quadratic program with linear equality, nonnegativity, and complementarity.
This video shows one of the major new feature in gurobi 9. Nonconvex quadratic programming dieter vandenbussche department of mechanical and industrial engineering university of illinois urbanachampaign workshop on integer programming and continuous optimization, 2004. Consequently, solution of such a problem that is, a nonconvex qp can take many orders of magnitude longer than the solution of a convex qp of comparable dimensions. Nocedal and wright, 2006 is also not appropriate for nonconvex problems. Global solutions of nonconvex standard quadratic programs. We introduce the suggestandimprove framework for general nonconvex quadratically constrained quadratic programs qcqps. An iterative method for nonconvex quadratically constrained. A decomposition method for nonconvex quadratically constrained quadratic programs chuangchuang sun and ran dai abstractthis paper examines the nonconvex quadratically constrained quadratic programming qcqp problems using a decomposition method.
Methods for convex and general quadratic programming. 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. We present a new heuristic for this problem, which enables one to obtain solutions. The complexity of the qp subproblem has been a major impediment to the formulation of secondderivative sqp methods although. A binarisation heuristic for nonconvex quadratic programming. Chuongs research was supported by the unsw vicechancellors postdoctoral research fellowship.
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