On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets

In the paper we prove that any nonconvex quadratic problem over some set ${K\subset \mathbb {R}^n}$ with additional linear and binary constraints can be rewritten as a linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem. This generalizes results obtained by Sturm and Zhang (Math Oper Res 28:246–267, 2003). Our result also generalizes the well-known completely positive representation result from Burer (Math Program 120:479–495, 2009), which is actually a special instance of our result with ${K=\mathbb{R}^n_{+}}$ .     

A note on set-semidefinite relaxations of nonconvex quadratic programs

We consider semidefinite, copositive, and more general, set-semidefinite programming relaxations of general nonconvex quadratic problems. For the semidefinite case a comparison between the feasible set of the original program and the feasible set of the relaxation has been given by Kojima and Tunçel (SIAM J Optim 10(3):750–778, 2000). In this paper the comparison is presented for set-positive relaxations which contain copositive relaxations as a special case.     

On the copositive representation of binary and continuous nonconvex quadratic programs

In this paper, we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of a small collection of NP-hard problems. A simplification, which reduces the dimension of the linear conic program, and an extension to complementarity constraints are established, and computational issues are discussed. Research partially supported by NSF Grant CCF-0545514.     

An LPCC approach to nonconvex quadratic programs

Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divided into two LPCCs: the first confirms whether the QP is bounded below on the feasible set and, if not, computes a feasible ray on which the QP is unbounded; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can be accomplished by a parameter-free, mixed integer-programming based, finitely terminating algorithm developed recently by the authors, which can be enhanced in this context by a new kind of valid cut derived from the second-order conditions of the QP and by exploiting the special structure of the LPCCs. Throughout, our treatment makes no boundedness assumption of the QP; this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption. The general theory is illustrated by 3 classes of indefinite problems: QPs with simple upper and lower bounds (existence of optimal solutions is guaranteed); same QPs with an additional inequality constraint (extending the case of simple bound constraints); and nonnegatively constrained copositive QPs (no guarantee of the existence of an optimal solution). We also present numerical results to support the special cuts obtained due to the QP connection.     

Hidden conic quadratic representation of some nonconvex quadratic optimization problems

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.     

The bounds of feasible space on constrained nonconvex quadratic programming

This paper presents a method to estimate the bounds of the radius of the feasible space for a class of constrained nonconvex quadratic programmings. Results show that one may compute a bound of the radius of the feasible space by a linear programming which is known to be a P$P$-problem [N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984) 373–395]. It is proposed that one applies this method for using the canonical dual transformation [D.Y. Gao, Canonical duality theory and solutions to constrained nonconvex quadratic programming, J. Global Optimization 29 (2004) 377–399] for solving a standard quadratic programming problem.     

A polyhedral study of nonconvex quadratic programs with box constraints

By reformulating quadratic programs using necessary optimality conditions, we obtain a linear program with complementarity constraints. For the case where the only constraints are bounds on the variables, we study a relaxation based on a subset of the optimality conditions. By characterizing its convex hull, we obtain a large class of valid inequalities. These inequalities are used in a branch-and-cut scheme, see [13].Mathematics Subject Classification (2000): 90C26, 90C27, 90C20     

Continuity of the feasible solution sets of probabilistic constrained programs

In this paper, we show the continuity of the feasibility set with respect to the reliability levels and with respect to the distribution of the random elements of a stochastic program with probabilistic constraints. Continuity is then used to obtain stability results for this type of stochastic program. An easy criterion is given for checking the conditions which guarantee the continuity of the feasibility set.The author wishes to thank Professor R. Wets for his comments.     

A Lagrangian decomposition approach to computing feasible solutions for quadratic binary programs

In this paper, we develop a Lagrangian decomposition based heuristic method for general quadratic binary programs (QBPs) with linear constraints. We extend the idea of Lagrangian decomposition by Chardaire and Sutter (Manag Sci 41(4):704–712, 1995) and Billionnet and Soutif (Eur J Oper Res 157(3):565–575, 2004a, Inf J Comput 16(2):188–197, 2004b) in which the quadratic objective is converted to a bilinear function by introducing auxiliary variables to duplicate the original complicating variables in the problem. Instead of using linear constraints to assure the equity between the two types of decision variables, we introduce generalized quadratic constraints and relax them with Lagrangian multipliers. Instead of computing an upper bound for a maximization problem, we focus on lower bounding with Lagrangian decomposition based heuristic. We take advantage of the decomposability presented in the Lagrangian subproblems to speed up the heuristic and identify one feasible solution at each iteration of the subgradient optimization procedure. With numerical studies on several classes of representative QBPs, we investigate the sensitivity of lower-bounding performance on parameters of the additional quadratic constraints. We also demonstrate the potentially improved quality of preprocessing in comparison with the use of a QBP solver.     

An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

     

On constraint sets of infinite linear programs over ordered fields

This paper provides a characterization of extreme points and extreme directions of the subsets of the space of generalized finite sequences occurring as the constraint sets of semi-infinite or infinite linear programs. The main result is that these sets are generated by (possibly infinitely many) extreme points and extreme directions. All results are valid over arbitrary ordered fields.     

Correction to: Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

Mathematical Programming - A correction to this paper has been published: https://doi.org/10.1007/s10107-021-01684-5     

ON THE MINIMUM FEASIBLE GRAPH FOR FOUR SETS

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Necessary and sufficient condition for local minima of a class of nonconvex quadratic programs

The author (1992, 1993) earlier studied the equivalence of a class of 0–1 quadratic programs and their relaxed problems. Thus, a class of combinatorial optimization problems can be solved by solving a class of nonconvex quadratic programs. In this paper, a necessary and sufficient condition for local minima of this class of nonconvex quadratic programs is given; this will be the foundation for study of algorithms.Research supported by Huo Yingdong Educational Foundation '93.     

On a class of quadratic programs

This paper considers a class of quadratic programs where the constraints ae linear and the objective is a product of two linear functions. Assuming the two linear factors to be non-negative, maximization and minimization cases are considered. Each case is analyzed with the help of a bicriteria linear program obtained by replacing the quadratic objective with the two linear functions. Global minimum (maximum) is attained at an efficient extreme point (efficient point) of the feasible set in the solution space and corresponds to an efficient extreme point (efficient point) of the feasible set in the bicriteria space. Utilizing this fact and certain other properties, two finite algorithms, including validations are given for solving the respective problems. Each of these, essentially, consists of solving a sequence of linear programs. Finally, a method is provided for relaxing the non-negativity assumption on the two linear factors of the objective function.     

On the representation of integers by quadratic forms

Let n 4 and let Q [X₁, ..., X_n] be a non-singular quadraticform. When Q is indefinite we provide new upper bounds for theleast non-trivial integral solution to the equation Q = 0, andwhen Q is positive definite we provide improved upper boundsfor the greatest positive integer k for which the equation Q= k is insoluble in integers, despite being soluble modulo everyprime power.     

On the representation of partially ordered sets

It is well known that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, ?) such that the supremum and the infimum of any finite setF ofP correspond, respectively to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomophic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI.