scholarly journals A New Spatial Branch and Bound Algorithm for Quadratic Program with One Quadratic Constraint and Linear Constraints

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jing Zhou
Keyword(s):  
Branch And Bound ◽  
High Efficiency ◽  
Real Life ◽  
Linear Constraints ◽  
Branch And Bound Algorithm ◽  
Quadratic Program ◽  
Constraint Matrix ◽  
Sdp Relaxation ◽  
Quadratic Constraint ◽  
Spatial Branch And Bound

This paper proposes a novel second-order cone programming (SOCP) relaxation for a quadratic program with one quadratic constraint and several linear constraints (QCQP) that arises in various real-life fields. This new SOCP relaxation fully exploits the simultaneous matrix diagonalization technique which has become an attractive tool in the area of quadratic programming in the literature. We first demonstrate that the new SOCP relaxation is as tight as the semidefinite programming (SDP) relaxation for the QCQP when the objective matrix and constraint matrix are simultaneously diagonalizable. We further derive a spatial branch-and-bound algorithm based on the new SOCP relaxation in order to obtain the global optimal solution. Extensive numerical experiments are conducted between the new SOCP relaxation-based branch-and-bound algorithm and the SDP relaxation-based branch-and-bound algorithm. The computational results illustrate that the new SOCP relaxation achieves a good balance between the bound quality and computational efficiency and thus leads to a high-efficiency global algorithm.

scholarly journals A Deterministic Method for Solving the Sum of Linear Ratios Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhenping Wang ◽  
Yonghong Zhang ◽  
Paolo Manfredi
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Branch And Bound ◽  
Nonconvex Programming ◽  
Real Life ◽  
Branch And Bound Algorithm ◽  
Separation Technique ◽  
Linear Programming Relaxation ◽  
Deterministic Method ◽  
Linearization Technique

Since the sum of linear ratios problem (SLRP) has many applications in real life, for globally solving it, an efficient branch and bound algorithm is presented in this paper. By utilizing the characteristic of the problem (SLRP), we propose a convex separation technique and a two-part linearization technique, which can be used to generate a sequence of linear programming relaxation of the initial nonconvex programming problem. For improving the convergence speed of this algorithm, a deleting rule is presented. The convergence of this algorithm is established, and some experiments are reported to show the feasibility and efficiency of the proposed algorithm.

scholarly journals A Branch and Bound Algorithm for Agile Earth Observation Satellite Scheduling

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaogeng Chu ◽  
Yuning Chen ◽  
Lining Xing
Keyword(s):  
Branch And Bound ◽  
Real Life ◽  
Engineering Application ◽  
Search Space ◽  
Branch And Bound Algorithm ◽  
Temporal Constraint ◽  
Look Ahead ◽  
Combined Use ◽  
Earth Observation Satellite ◽  
Satellite Scheduling

The agile earth observing satellite scheduling (AEOSS) problem consists of scheduling a subset of images among a set of candidates that satisfy imperative constraints and maximize a gain function. In this paper, we consider a new AEOSS model which integrates a time-dependent temporal constraint. To solve this problem, we propose a highly efficient branch and bound algorithm whose effective ingredients include a look-ahead construction method (for generating a high quality initial lower bound) and a combined use of three pruning strategies (which help to prune a large portion of the search space). We conducted computational experiments on a set of test data that were generated with information from real-life scenarios. The results showed that the proposed algorithm is efficient enough for engineering application. In particular, it is able to solve instances with 55 targets to optimality within 164 seconds on average. Furthermore, we carried out additional experiments to analyze the contribution of each key algorithm ingredient.

scholarly journals Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems

2021 ◽  
Author(s):  
G. Liuzzi ◽  
M. Locatelli ◽  
V. Piccialli ◽  
S. Rass
Keyword(s):  
Game Theory ◽  
Branch And Bound ◽  
Switching Costs ◽  
Computational Cost ◽  
Linear Constraints ◽  
Point Of View ◽  
Quadratic Term ◽  
Mixed Strategies ◽  
Computational Point ◽  
Spatial Branch And Bound

AbstractIn this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as and , are unable to solve to optimality even test instances with $$n=50$$ n = 50 variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

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