Concave Quadratic Programming Research Articles

Active Fault Diagnosis for Stochastic Systems Within Bayesian Minimum Risk Decision Framework

In this paper, the active fault diagnosis (AFD) problem is investigated for a class of discrete-time stochastic systems. We consider multiple fault modes of the system, and the different types of misdiagnoses of these faults lead to different losses. A novel AFD framework is developed based on the Bayesian minimum risk decision framework. The misdiagnosis risk is derived to characterize the total losses caused by the misdiagnoses of all considered faults. The input is specifically designed to improve diagnosis performance by minimizing the risk of misdiagnosis. Since it is intractable to express the misdiagnosis risk in the closed form, we deduce an upper bound on it to establish a novel input design criterion based on the Bhattacharyya distance. The optimal input signal can be obtained for AFD via concave quadratic programming. For diagnosis performance enhancement and computation simplification purposes, we propose two new semi-online implementation manners of the input design for stochastic systems with and without multi-fault modes switching respectively. Given the optimal input signal and the proposed implementation manner, faults are diagnosed based on the minimum misdiagnosis risk decision principle. The effectiveness and superiority of the proposed AFD method are demonstrated by simulation results on a four-tank system.

Using concave optimization methods for inexact quadratic programming problems with an application to waste management

Quadratic programming is potentially capable of strategic decision making in real world problems. However, practical problems rarely conform to crisp parameters, and hence the prospects of these problems with inexact parameters are inevitably higher. The existing studies regarding public welfare schemes/ organizations reveal that their objectives end up as minimization of cost functions and are governed by linear or concave quadratic programming problems. The present study proposes a method that can be applied to concave type quadratic objective function subject to linear constraints with inexact parameters. A comparison is also drawn with existing methods to establish its simplicity and efficiency. Further, a numerical example is illustrated, and finally, a waste management problem is formulated and solved using the proposed method.

A successive linear approximation algorithm for the global minimization of a concave quadratic program

In this work, we propose an algorithm for finding an approximate global minimum of a concave quadratic function with a negative semi-definite matrix, subject to linear equality and inequality constraints, where the variables are bounded with finite or infinite bounds. The proposed algorithm starts with an initial extreme point, then it moves from the current extreme point to a new one with a better objective function value. The passage from one basic feasible solution to a new one is done by the construction of certain approximation sets and solving a sequence of linear programming problems. In order to compare our algorithm with the existing approaches, we have developed an implementation with MATLAB and conducted numerical experiments on numerous collections of test problems. The obtained numerical results show the accuracy and the efficiency of our approach.

A new algorithm for concave quadratic programming

The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a modified version by restricting the feasible set. This leads to a new bound for quadratic programs. We demonstrate that the dual of the bound is a semi-definite relaxation of quadratic programs. In addition, we probe the relationship between this bound and the well-known bounds in the literature. In the second part, thanks to the new bound, we propose a branch and cut algorithm for concave quadratic programs. We establish that the algorithm enjoys global convergence. The effectiveness of the method is illustrated for numerical problem instances.

Energy-maximising control of wave energy converters using a moment-domain representation

Wave Energy Converters (WECs) have to be controlled to ensure maximum energy extraction from waves while considering, at the same time, physical constraints on the motion of the real device and actuator characteristics. Since the control objective for WECs deviates significantly from the traditional reference “tracking” problem in classical control, the specification of an optimal control law, that optimises energy absorption under different sea-states, is non-trivial. Different approaches based on optimal control methodologies have been proposed for this energy-maximising objective, with considerable diversity on the optimisation formulation. Recently, a novel mathematical tool to compute the steady-state response of a system has been proposed: the moment-based phasor transform. This mathematical framework is inspired by the theory of model reduction by moment-matching and considers both continuous and discontinuous inputs, depicting an efficient and closed-form method to compute such a steady-state behaviour. This study approaches the design of an energy-maximising optimal controller for a single WEC device by employing the moment-based phasor transform, describing a pioneering application of this novel moment-matching mathematical scheme to an optimal control problem. Under this framework, the energy-maximising optimal control formulation is shown to be a strictly concave quadratic program, allowing the application of well-known efficient real-time algorithms.

Decomposition Procedure for Solving NLP and QP Problems based on Lagrange and Sander's Method

This paper develops a decompose procedure for finding the optimal solution of convex and concave Quadratic Programming (QP) problems together with general Non-linear Programming (NLP) problems. The paper also develops a sophisticated computer technique corresponding to the author's algorithm using programming language MATHEMATICA. As for auxiliary by making comparison, the author introduces a computer-oriented technique of the traditional Karush-Kuhn-Tucker (KKT) method and Lagrange method for solving NLP problems. He then modify the Sander's algorithm and develop a new computational technique to evaluate the performance of the Sander's algorithm for solving NLP problems. The author observe that the technique avoids some certain numerical difficulties in NLP and QP. He illustrates a number of numerical examples to demonstrate his method and the modified algorithm.

An Algorithmic Technique for Solving Non-Linear Programming and Quadratic Programming Problems

In this paper, we improve a combined algorithm and develop a uniform computer technique for solving constrained, unconstrained Non Linear Programming (NLP) and Quadratic Programming (QP) problems into a single framework. For this, we first review the basic algorithms of convex and concave QP as well as general NLP problems. We also focus on the development of the graphical representations. We use MATHEMATICA 9.0 to develop this algorithmic technique. We present a number of numerical examples to demonstrate our method.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 41-55

Calibration, sharpness and the weighting of experts in a linear opinion pool

Linear opinion pools are the most common form of aggregating the probabilistic judgments of multiple experts. Here, the performance of such an aggregation is examined in terms of the calibration and sharpness of the component judgments. The performance is measured through the average quadratic score of the aggregate. Trade-offs between calibration and sharpness are examined and an expression for the optimal weighting of two dependent experts in a linear combination is given. Circumstances where one expert would be disqualified are investigated. Optimal weights for the multiple, dependent experts are found through a concave quadratic program.

On Box Constrained Concave Quadratic Optimization

This paper presents a class of differential flows to solve concave quadratic programming problems under box constraints. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. It is shown how the differential flow reaches a vertex point of the box, leading to a global minimizer of the concave quadratic programming. Some illustrative examples are also presented.

Spherical coverage verification

We consider the problem of covering hypersphere by a set of spherical hypercaps. This sort of problem has numerous practical applications such as error correcting codes and reverse k-nearest neighbor problem. Using the reduction of non-degenerated concave quadratic programming (QP) problem, we demonstrate that spherical coverage verification is NP hard. We propose a recursive algorithm based on reducing the problem to several lower dimension subproblems. We test the performance of the proposed algorithm on a number of generated constellations. We demonstrate that the proposed algorithm, in spite of its exponential worst-case complexity, is applicable in practice. In contrast, our results indicate that spherical coverage verification using QP solvers that utilize heuristics, due to numerical instability, may produce false positives.

Several multi-criteria programming methods for classification

In this paper, we propose a general optimization-based model for classification. Then we show that some well-known optimization-based methods for classification, which were developed by Shi et al. [Data mining in credit card portfolio management: a multiple criteria decision making approac. In: Koksalan M, Zionts S, editors. Multiple criteria decision making in the new millennium. Berlin: Springer; 2001. p. 427–36] and Freed and Glover [A linear programming approach to the discriminant problem. Decision Sciences 1981; 12: 68–79; Simple but powerful goal programming models for discriminant problems. European Journal of Operational Research 1981; 7: 44–60], are special cases of our model. Moreover, three new models, MCQP (multi-criteria indefinite quadratic programming), MCCQP (multi-criteria concave quadratic programming) and MCVQP (multi-criteria convex programming), are developed based on the general model. We also propose algorithms for MCQP and MCCQP, respectively. Then we apply these models to three real-life problems: credit card accounts, VIP mail-box and social endowment insurance classification. Extensive experiments are done to compare the efficiency of these methods.

An Efficient Algorithm for Capacitated Multifacility Location Problems

In this paper, a squared-Euclidean distance multifacility location problem with inseparable demands under balanced transportation constraints is analyzed. Using calculus to project the problem onto the space of allocation variables, the problem becomes minimizing concave quadratic integer programming problem. The algorithm based on extreme point ranking method combining with logical techniques is developed. The numerical experiments are randomly generated to test efficiency of the proposed algorithm compared with a linearization algorithm. The results show that the proposed algorithm provides a better solution on average with less processing time for all various sizes of problems.

Global minimization algorithms for concave quadratic programming problems

: In this article, we consider the concave quadratic programming problem which is known to be NP hard. Based on the improved global optimality conditions by [Dür, M., Horst, R. and Locatelli, M., 1998, Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications, 217, 637–649] and [Hiriart-Urruty, J.B. and Ledyav, J.S., 1996, A note in the characterization of the global maxima of a convex function over a convex set, Journal of Convex Analysis, 3, 55–61], we develop a new approach for solving concave quadratic programming problems. The main idea of the algorithms is to generate a sequence of local minimizers either ending at a global optimal solution or at an approximate global optimal solution within a finite number of iterations. At each iteration of the algorithms we solve a number of linear programming problems with the same constraints of the original problem. We also present the convergence properties of the proposed algorithms under some conditions. The efficiency of the algorithms has been demonstrated with some numerical examples.

Control-relevant design of periodic test input signals for iterative open-loop identification of multivariable FIR systems

In this paper, the problem of control-relevant test input design for iterative system identification is considered. Two approaches are presented for the L-optimal input test signal design by Cooley and Lee (2001). which minimizes the a posteriori closed-loop error based on a priori information given by the data already collected from previous iterations. To simplify the computation, we limit our design to a period signal of period n, which is the number of FIR coefficients. The first approach is direct time-domain search for optimal test input values, which can be formulated as a nonconvex optimization or a concave Quadratic Program. The second approach considers frequency domain solution of the L-optimal design problem to obtain the optimal discrete spectra of the test signals, which is subsequently realized as a periodic signal.

A note on composite concave quadratic programming

In this paper we present a pivotal-based algorithm for the global minimization of composite concave quadratic functions subject to linear constraints. It is shown that certain subclasses of this family yield easy-to-solve line search subproblems. Since the proposed algorithm is equivalent in efficiency to a standard parametric complementary pivoting procedure, the implication is that conventional parametric quadratic programming algorithms can now be used as tools for the solution of much wider class of complex global optimization problems.

Cutting Plane/Tabu Search Algorithms for Low Rank Concave Quadratic Programming Problems

In this paper, we will propose an efficient heuristic algorithm for solving concave quadratic programming problems whose rank of the objective function is relatively small. This algorithm is a combination of Tuy‘s cutting plane to eliminate the feasible region and a kind of tabu-search method to find a ’good‘ vertex. We first generate a set of V of vertices and select one of these vertices as a starting point at each step, and apply tabu-search and Tuy‘s cutting plane algorithm where the list of tabu consists of those vertices eliminated by cutting planes and those newly generated vertices by cutting planes. When all vertices of the set V are eliminated, the algorithm is terminated. This algorithm need not converge to a global minimum, but it can work very well when the rank is relatively small (up to seven). The incumbent solutions are in fact globally optimal for all tested problems. We also propose an alternative algorithm by incorporating Rosen‘s hyperrectangle cut. This algorithm is more efficient than the combination of Tuy‘s cutting plane and tabu-search.

A low-rank bilinear programming approach for sub-optimal solution of the quadratic assignment problem

This paper is concerned with a new approach for solving quadratic assignment problems (QAP). We first reformulate QAP as a concave quadratic programming problem and apply an outer approximation algorithm. In addition, an improvement routine is incorporated in the final stage of the algorithm. Computational experiments on a set of standard data demonstrate that this algorithm can yield favorable results with a relatively low computational effort.

Decomposition approach for the global minimization of biconcave functions over polytopes

A decomposition approach is proposed for minimizing biconcave functions over polytopes. Important special cases include concave minimization, bilinear and indefinite quadratic programming for which new algorithms result. The approach introduces a new polyhedral partition and combines branch-and-bound techniques, outer approximation, and projection of polytopes in a suitable way.

A Global Optimization Algorithm for Concave Quadratic Programming Problems

Using a global optimality criterion for concave quadratic problems due to Hiriart–Urruty and Lemaréchal, the authors present an algorithm which manages to “escape” from a local solution and lead towards the global one. In addition, this procedure recognizes the unsolvability of the problem (due to unboundedness of the objective function on the feasible region) and also generates an improving feasible direction even if the starting point does not satisfy the Karush/Kuhn/Tucker conditions. As a key subroutine, a recursive procedure will be used which determines whether or not a given symmetric $n \times n$-matrix is copositive, i.e., yields a quadratic form that is positive on a given polyhedral cone. Both this subroutine and the main body of the algorithm frequently employ the simplex method, while all other operations are elementary.

Using copositivity for global optimality criteria in concave quadratic programming problems

In this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems. Using this condition, it follows that, from the perspective of worst-case complexity of concave quadratic problems, the difference between local and global optimality conditions is not as large as in general. As an essential ingredient, we here use thee-subdifferential calculus via an approach of Hiriart-Urruty and Lemarechal (1990).