Indefinite Quadratic Programming Research Articles

Solving integer indefinite quadratic bilevel programs with multiple objectives at the upper level

Solving integer indefinite quadratic bilevel programs with multiple objectives at the upper level

Retracted: Branch-and-Reduction Algorithm for Indefinite Quadratic Programming Problem

Retracted: Branch-and-Reduction Algorithm for Indefinite Quadratic Programming Problem

DC algorithms in Hilbert spaces and the solution of indefinite infinite-dimensional quadratic programs

DC algorithms in Hilbert spaces and the solution of indefinite infinite-dimensional quadratic programs

Fuzzy goal programming technique for multi-objective indefinite quadratic bilevel programming problem

Fuzzy goal programming technique for multi-objective indefinite quadratic bilevel programming problem

An approximation algorithm for indefinite mixed integer quadratic programming

In this paper, we give an algorithm that finds an \(\epsilon \)-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices.

On a solution method in indefinite quadratic programming under linear constraints

We establish some properties of the Proximal Difference-of-Convex functions decomposition algorithm in indefinite quadratic programming under linear constraints. The first property states that any iterative sequence generated by the algorithm is root linearly convergent to a Karush–Kuhn–Tucker point, provided that the problem has a solution. The second property says that iterative sequences generated by the algorithm converge to a locally unique solution of the problem if the initial points are taken from a suitably chosen neighbourhood of it. Through a series of numerical tests, we analyse the influence of the decomposition parameter on the rate of convergence of the iterative sequences and compare the performance of the Proximal Difference-of-Convex functions decomposition algorithm with that of the Projection Difference-of-Convex functions decomposition algorithm. In addition, the performances of the above algorithms and the Gurobi software in solving some randomly generated nonconvex quadratic programs are compared.

Semisupervised Fuzzy Clustering With Fuzzy Pairwise Constraints

In semisupervised fuzzy clustering, this article extends the traditional pairwise constraint (i.e., must-link or cannot-link) to fuzzy pairwise constraint. The fuzzy pairwise constraint allows a supervisor to provide the grade of similarity or dissimilarity between the implicit fuzzy vectors of a pair of samples. This constraint can represent more complicated relationship between the pair of samples and avoid eliminating the fuzzy characteristics. Then, we propose a semisupervised fuzzy clustering with fuzzy pairwise constraints (SSFPC). The nonconvex optimization problem in our SSFPC is solved by a modified expectation-maximization algorithm, involving to solve several indefinite quadratic programming problems (IQPPs). Further, a diagonal block coordinate decent (DBCD) algorithm is proposed for these IQPPs, whose stationary points are guaranteed, and the global solutions can be obtained under certain conditions. To suit for different applications, the SSFPC is extended into various metric spaces, e.g., the reproducing kernel Hilbert space. Experimental results on several benchmark datasets and a facial expression database demonstrate the outperformance of our SSFPC compared with some state-of-the-art clustering models

Branch and cut method for solving integer indefinite quadratic bilevel programs

Branch and cut method for solving integer indefinite quadratic bilevel programs

Branch-and-Reduction Algorithm for Indefinite Quadratic Programming Problem

This paper presents a rectangular branch-and-reduction algorithm for globally solving indefinite quadratic programming problem (IQPP), which has a wide application in engineering design and optimization. In this algorithm, first of all, we convert the IQPP into an equivalent bilinear optimization problem (EBOP). Next, a novel linearizing technique is presented for deriving the linear relaxation programs problem (LRPP) of the EBOP, which can be used to obtain the lower bound of the global optimal value to the EBOP. To obtain a global optimal solution of the EBOP, the main computational task of the proposed algorithm involves the solutions of a sequence of LRPP. Moreover, the global convergent property of the algorithm is proved, and numerical experiments demonstrate the higher computational performance of the algorithm.

Integer programming in parameterized complexity: Five miniatures

Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra’s algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k)poly(n). We focus on: •Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.•Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.•Minding the poly(n): reducing f(k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds. Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, Max-q-Cut, and certain other coloring problems by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, indefinite quadratic programs in fixed dimension, parametric integer programs in fixed dimension, and 2-stage stochastic integer programs.

An algorithm for piece-wise indefinite quadratic programming problem

An indefinite quadratic programming problem is a mathematical programming problem which is a product of two linear factors. In this paper, the piece-wise indefinite quadratic programming problem (PIQPP) is considered. Here, the objective function is a product of two continuous piecewise linear functions defined on a non-empty and compact feasible region. In the present paper, the optimality criterion is derived and explained in order to solve PIQPP. While solving PIQPP, we will come across certain variables which will not satisfy the optimality condition. For these variables, cases have been elaborated so as to move from one basic feasible solution to another till we reach the optimality. An algorithmic approach is proposed and discussed for the pr PIQPP problem. A numerical example is presented to decipher the tendered method.

A note on solving multi-objective integer indefinite quadratic fractional programs

In this note we have discussed that a simplex like algorithm to solve a indefinite quadratic fractional programming problem proposed by Mekhilef et al. (Ann Oper Res, 2019. https://doi.org/10.1007/s10479-019-03178-2) fails to find its optimal solution and so it may not generate the actual set of efficient points of the corresponding multi-objective integer indefinite quadratic fractional programs. A counter example in support of this argument is also given.

Globally Solving Nonconvex Quadratic Programs via Linear Integer Programming Techniques

We reformulate a (indefinite) quadratic program (QP) as a mixed-integer linear programming (MILP) problem by first reformulating a QP as a linear complementary problem, and then using binary variables and big-M constraints to model its complementary constraints. To obtain such reformulation, we use fundamental results on the solution of perturbed linear systems to impose bounds on the QP’s dual variables without eliminating any of its (globally) optimal primal solutions. Reformulating a nonconvex QP as a MILP problem allows the use of current state-of-the-art MILP solvers to find its global optimal solution. To illustrate this, we compare the performance of this MILP-based solution approach, labeled quadprogIP, with quadprogBB, BARON, and CPLEX. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. Also, unlike quadprogBB, quadprogIP is able to solve QP instances in which the dual feasible set is unbounded. The MATLAB code quadprogIP and the instances used to perform the reported numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP .

An exact method for solving multi-objective integer indefinite quadratic programs

ABSTRACTThis article tackles the optimization of a multi-objective integer indefinite quadratic maximization problem. The proposed algorithm for this class of nonlinear multi-objective problems obtains the whole set of efficient solutions by a Branch-and-Bound method with cuts. At every node of the branching tree, a relaxed problem, where both integrality and efficiency constraints are dropped, is solved by the simplex method. By tests and fathoming rules the algorithm decides, whether the node is fathomed, the relaxed problem is branched into subproblems or cuts are added. After the presentation and validation of the algorithm, a numerical example and computational experiments are provided.

Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 2: Robinson Stability

In Part 1 of this paper, we have estimated the Fr\'echet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given.

MINQ8: general definite and bound constrained indefinite quadratic programming

We propose new algorithms for (i) the local optimization of bound constrained quadratic programs, (ii) the solution of general definite quadratic programs, and (iii) finding either a point satisfying given linear equations and inequalities or a certificate of infeasibility. The algorithms are implemented in Matlab and tested against state-of-the-art quadratic programming software.

D.C. programming approach for solving an applied ore-processing problem

This paper was motivated by a practical optimization problem formulated at the Erdenet Mining Corporation (Mongolia). By solving an identification problem for a chosen design of experiment we developed a quadratic model that quite adequately represents the experimental data. The problem obtained turned out to be the indefinite quadratic program, which we solved by applying the global search theory for a d.c. programming developed by A.S. Strekalovsky [13]-[15]. According to this d.c. optimization theory, we performed a local search that takes into account the structure of the problem in question, and constructed procedures of escaping critical points provided by the local search. The algorithms proposed for d.c. programming were verified using a set of test problems as well as a copper content maximization problem arising at the mining factory.

Cooperative MIMO relay beamforming in overloaded DS/CDMA multicast networks

A new beamforming scheme is proposed for multi-input multi-output (MIMO) non-regenerative relays in an overloaded synchronous direct sequence code division multiple access (DS-CDMA) network in which the number of users is larger than the processing gain. In this scheme, several transmitters multicast their data to their groups of receivers through MIMO relays and DS-CDMA technique is utilised to decrease the level of multiple access interference (MAI) at the relays and the receivers. At the relays, the data of transmitters are despread, beamformed and re-spread. The aim of the beamformer is to minimise the total relays transmission power subject to the signal to interference and noise ratio (SINR) constraints at the receivers. This optimisation problem is an indefinite quadratic (non-convex) programming. Semi-definite programming (SDP) and second-order cone programming (SOCP) are two existing methods to solve the problem. However, they, respectively, suffer from high computational complexity and performance degradation when the number of antennas or users increase. To effectively mitigate these deficiencies, we propose a novel DC (difference of two convex functions) programming method. Our simulation results demonstrate the computational advantage of the proposed DC programming over SDP, while a better performance over SOCP is achieved. Moreover, the superiority of the proposed MIMO-CDMA scheme over other multiplexing schemes is shown by several scenarios.

A Sparsity Preserving Convexification Procedure for Indefinite Quadratic Programs Arising in Direct Optimal Control

Quadratic programs (QP) with an indefinite Hessian matrix arise naturally in some direct optimal control methods, e.g., as subproblems in a sequential quadratic programming scheme. Typically, the Hessian is approximated with a positive definite matrix to ensure having a unique solution; such a procedure is called regularization. We present a novel regularization method tailored for QPs with optimal control structure. Our approach exhibits three main advantages. First, when the QP satisfies a second order sufficient condition for optimality, the primal solution of the original and the regularized problem are equal. In addition, the algorithm recovers the dual solution in a convenient way. Second, and more importantly, the regularized Hessian bears the same sparsity structure as the original one. This allows for the use of efficient structure-exploiting QP solvers. As a third advantage, the regularization can be performed with a computational complexity that scales linearly in the length of the control horizon. We showcase the properties of our regularization algorithm on a numerical example for nonlinear optimal control. The results are compared to other sparsity preserving regularization methods.

Tight bounds on indefinite separable singly-constrained quadratic programs in linear-time

Non-convex knapsack separable quadratic optimization with compact box constraints is an NP-hard problem. We present tight lower and upper bounding procedures that are computationally-efficient as the problem size grows. The lower bound is based on Lagrangian relaxation, and it is computed in linear-time. When the bound is not an exact global solution, a worst-case bound-quality measure is developed. Moreover, the lower bounding (LB) solution is improved to construct a feasible solution, leading to an upper bound (UB) on the given problem. The UB is based on fixing the convex variables and a subset of non-convex and linear variables at the LB solution, and considering the remaining problem, which is always feasible. The UB is computable with linear-time complexity, and it is the global optimum in certain verifiable cases when the duality gap is zero. In our limited computational experiments, the UB has very small relative gap with an exact global optimum solution when there is a nonzero duality gap. Performance of the bounds is demonstrated through a broad range of randomly generated problem instances and comparisons with existing global and non-global solvers. The proposed method on these indefinite problems of extremely large size is an order of magnitude faster than the alternative solvers, including IBM's commercial software Cplex12.6.