Additional Quadratic Constraints Research Articles

On the exactness of a simple relaxation for the extended Celis–Dennis–Tapia subproblem

In this work, we study the so-called extended Celis–Dennis–Tapia subproblem that are classical trust region subproblems with an additional quadratic constraint. For this class of problems, we propose a family of relaxations, and study some of its key properties, for instance, convexity issues, exactness of the relaxation, sufficient condition for exactness. The relaxation proposed is obtained adapting to our case a methodology recently developed for trust region problems with an additional conic constraint. We also find sufficient conditions for exactness of the proposed relaxation and discuss the relations between them. Finally, some numerical experiments are presented for testing the proposed relaxation and the sufficient conditions for its tightness.

Global solution of non-convex quadratically constrained quadratic programs

The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let be a QCQP. Our approach to solve is first to build an equivalent mixed-integer quadratic problem . This equivalent problem has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints , where x are the initial variables of problem . We then propose to solve by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON andGloMIQO.

Optimal Network Control With Stochastic Link Variations and Indefinite Quadratic Constraints

In this paper, we consider the problem of optimal control of a network with stochastic links modeled as multiplicative noise with given second moments and arbitrary probability distributions. Additional quadratic constraints are imposed, where the quadratic forms may be indefinite and, thus, not necessarily convex. We show that the problem can be transformed to a semidefinite program. The optimization problem is over a certain variable serving as the covariance matrix of the state and the controller. We show that affine controllers are optimal and depend on the optimal covariance matrix. Furthermore, we show that optimal controllers are linear if all of the quadratic forms are convex in the control variable. The solutions are presented for the finite and infinite horizon cases. We give a necessary and sufficient condition for mean square stabilizability of the dynamical system with additive and multiplicative noise. The condition is a Lyapunov-like condition whose solution is again given by the covariance matrix of the state and the control variable. The results are illustrated with an example.

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.

Adaptive Nullforming to Mitigate Ground Clutter on the National Weather Radar Testbed Phased Array Radar

With the decreasing cost of phased array antennas, their use for weather surveillance is becoming more practical. A significant advantage of phased arrays that can be applied to weather surveillance is spatial filtering. Using adaptive nullforming to spatially filter clutter is a novel approach to clutter mitigation, which is not possible with conventional parabolic reflector antennas. Moreover, spatial filtering is also applicable to phased-array-specific techniques such as beam multiplexing and adaptive scanning when only a few pulses are available for processing; this situation is particularly challenging for conventional ground clutter filters. The National Weather Radar Testbed Phased Array Radar (NWRT PAR) provides an opportunity to test some of these new capabilities. In this paper, a linearly constrained minimum power algorithm with an additional quadratic constraint is applied to weather data collected using the NWRT PAR and its multichannel receiver. Both the original algorithm and a recursive least squares version are utilized to show reflectivity and velocity data where both weather and ground clutter are present. Doppler spectra from selected range gates are examined to illustrate the performance of adaptive nullforming. Issues such as the number of samples needed to estimate the covariance matrix are explored. As far as we know, this is the first time that these types of techniques have been used to mitigate ground clutter contamination on a weather surveillance radar.

Projection-based robust adaptive beamforming with quadratic constraint

A new robust adaptive beamforming technique is proposed in this study to address performance degradation of adaptive beamforming methods in the presence of steering vector mismatch. Actual steering vector of desired signal is estimated by solving a convex optimization problem with the objective constructed by minimizing the sum of estimated desired steering vector projections onto noise eigenvectors. The beamformer performs well at high signal-to-noise ratio (SNR) with the orthogonality between presumed desired steering vector and mismatch vector as a single constraint. Feasibility and necessity of adding an additional quadratic constraint are verified through detailed performance analysis with random matrix theory, improving the performance at low SNR. The parameter determination approach is provided to allow the proposed beamformer to function properly in practical situations. Both the theoretical analysis and simulation results demonstrate the proposed method is robust against any steering vector mismatch.

Linear and parabolic relaxations for quadratic constraints

This paper presents new techniques for filtering boxes in the presence of an additional quadratic constraint, a problem relevant for branch and bound methods for global optimization and constraint satisfaction. This is done by generating powerful linear and parabolic relaxations from a quadratic constraint and bound constraints, which are then subject to standard constraint propagation techniques. The techniques are often applicable even if the original box is unbounded in some but not all variables. As an auxiliary tool--needed to make our theoretical results implementable in floating-point arithmetic without sacrificing mathematical rigor--we extend the directed Cholesky factorization from Domes and Neumaier (SIAM J Matrix Anal Appl 32:262---285, 2011) to a partial directed Cholesky factorization with pivoting. If the quadratic constraint is convex and the initial bounds are sufficiently wide, the final relaxation and the enclosure are optimal up to rounding errors. Numerical tests show the usefulness of the new factorization methods in the context of filtering.

A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation

Let $$(MQP)$$ be a general mixed-integer quadratic program that consists of minimizing a quadratic function $$f(x) = x^TQx +c^Tx$$ subject to linear constraints. Our approach to solve $$(MQP)$$ is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables $$y_{ij}$$ , additional quadratic constraints $$y_{ij}=x_ix_j$$ , a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in Billionnet et al. (Math Program 131(1):381---401, 2012), the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints $$y_{ij}=x_ix_j$$ to solve $$(MQP)$$ . Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with the approach of Billionnet et al. (2012) and with the general mixed-integer nonlinear solver BARON (Sahinidis and Tawarmalani, Global optimization of mixed-integer nonlinear programs, user's manual, 2010).

Fuzzy Data Decision Support in Portfolio Selection: a Possibilistic Safety-first Model

Vast pools of historical financial information are available on economies, industry, and individual companies that affect investors’ selection of appropriate portfolios. Fuzzy data provides a good tool to reflect investors’ opinions based on this information. A possibilistic mean variance safety-first portfolio selection model is developed to support investors’ decision making, to take into consideration this fuzzy information. The possibilistic-programming problem can be transformed into a linear optimal problem with an additional quadratic constraint using possibilistic theory. We propose a cutting plane algorithm to solve the programming problem. A numerical example is given to illustrate our approach.

The very-extended trombone

Starting from the very-extended Kac-Moody algebra $E_{11}$, we consider the algebra $E_{11,D}^{local}$, obtained by adding to the non-negative level $E_{11}$ generators the $D$-dimensional momentum operator and an infinite set of additional generators that promote the global $E_{11}$ symmetries to gauge ones. We determine all the possible trombone deformations of this algebra, that is the deformations that involve the $D$-dimensional scaling operator. The Jacobi identities imply that such deformations are uniquely determined by a single tensor belonging to the same representation of the internal symmetry group as the vector generators and satisfying additional quadratic constraints. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible maximal supergravity theory in which the global scaling symmetry is gauged in any dimension. All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories. The tensor that parametrises the deformation is identified with the embedding tensor that is used to parametrise all maximal supergravity theories with gauged scaling symmetry, and the quadratic constraints that we determine exactly coincide with the field theory results.

A cutting plane algorithm for MV portfolio selection model

This paper deals with a portfolio selection problem with fuzzy return rates. A possibilistic mean variance (FMVC) portfolio selection model was proposed. The possibilistic programming problem can be transformed into a linear optimal problem with an additional quadratic constraint by possibilistic theory. For such problems there are no special standard algorithms. We propose a cutting plane algorithm to solve (FMVC). The nonlinear programming problem can be solved by sequence linear programming problem. A numerical example is given to illustrate the behavior of the proposed model and algorithm.

A New Ship Scheduling Set Packing Model Considering Limited Risk

In this paper, we propose a new ship scheduling set packing model considering limited risk or variance. The set packing model is used in many applications, such as vehicle routing, crew scheduling, ship scheduling, cutting stock and so on. As long as the ship scheduling is concerned, there exits many unknown external factors such as machine breakdown, climate change and transportation cost fluctuation. However, existing ship scheduling models have not considered those factors apparently. We use a quadratic set packing model to limit the variance of expected cost of ship scheduling problems under stochastic spot rates. Set problems are NP-complete, and additional quadratic constraint makes the problems much harder. We implement Kelley's cutting plane method to replace the hard quadratic constraint by many linear constrains and use branch-and-bound algorithm to get the optimal integral solution. Some meaningful computational results and comments are provided.