scholarly journals An effective algorithm for globally solving quadratic programs using parametric linearization technique

2018 ◽  
Vol 16 (1) ◽  
pp. 1300-1312
Author(s):  
Shuai Tang ◽  
Yuzhen Chen ◽  
Yunrui Guo
Keyword(s):  
Linear Programming ◽  
Optimal Solution ◽  
Route Optimization ◽  
Linear Programming Relaxation ◽  
Effective Algorithm ◽  
Quadratic Programs ◽  
Parametric Linear Programming ◽  
Quadratic Constraints ◽  
Linearization Technique ◽  
Relaxation Problem

AbstractIn this paper, we present an effective algorithm for globally solving quadratic programs with quadratic constraints, which has wide application in engineering design, engineering optimization, route optimization, etc. By utilizing new parametric linearization technique, we can derive the parametric linear programming relaxation problem of the quadratic programs with quadratic constraints. To improve the computational speed of the proposed algorithm, some interval reduction operations are used to compress the investigated interval. By subsequently partitioning the initial box and solving a sequence of parametric linear programming relaxation problems the proposed algorithm is convergent to the global optimal solution of the initial problem. Finally, compared with some known algorithms, numerical experimental results demonstrate that the proposed algorithm has higher computational efficiency.

scholarly journals An Effective Global Optimization Algorithm for Quadratic Programs with Quadratic Constraints

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 424
Author(s):  
Dongwei Shi ◽  
Jingben Yin ◽  
Chunyang Bai
Keyword(s):  
Numerical Experiments ◽  
Optimal Solution ◽  
Linearization Method ◽  
Linear Programming Relaxation ◽  
Global Optimization Algorithm ◽  
Quadratic Programs ◽  
Global Optimal Solution ◽  
Quadratic Constraints ◽  
Relaxation Problem ◽  
Global Optimal

This paper will present an effective algorithm for globally solving quadratic programs with quadratic constraints. In this algorithm, we propose a new linearization method for establishing the linear programming relaxation problem of quadratic programs with quadratic constraints. The proposed algorithm converges with the global optimal solution of the initial problem, and numerical experiments show the computational efficiency of the proposed 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 Linear Programming Relaxation DEA Model for Selecting a Single Efficient Unit with Variable RTS Technology

2021 ◽  
Vol 12 (2) ◽  
pp. 131-137
Author(s):  
Reza Akhlaghi ◽  
Mohsen Rostamy-Malkhalifeh ◽  
Alireza Amirteimoori ◽  
Sohrab Kordrostami
Keyword(s):  
Decision Making ◽  
Linear Programming ◽  
Programming Model ◽  
Returns To Scale ◽  
Linear Programming Relaxation ◽  
Data Envelopment ◽  
Relaxation Model ◽  
Relaxation Problem ◽  
Lp Model ◽  
Variable Returns To Scale

The selection-based problem is a type of decision-making issue which involves opting for a single option among a set of available alternatives. In order to address the selection-based problem in data envelopment analysis (DEA), various integrated mixed binary linear programming (MBLP) models have been developed. Recently, an MBLP model has been proposed to select a unit in DEA with variable returns-to-scale technology. This paper suggests utilizing the linear programming relaxation model rather than the MBLP model. The MBLP model is proved here to be equivalent to its linear programming relaxation problem. To the best of the authors’ knowledge, this is the first linear programming model suggested for selecting a single efficient unit in DEA under the VRS (Variable Returns to Scale) assumption. Two theorems and a numerical example are provided to validate the proposed LP model from both theoretical and practical perspectives.

scholarly journals Inductive linearization for binary quadratic programs with linear constraints

4OR ◽  
2020 ◽  
Author(s):  
Sven Mallach
Keyword(s):  
Linear Programming ◽  
Integer Programming ◽  
Linear Constraints ◽  
General Purpose ◽  
Mixed Integer ◽  
Quadratic Programs ◽  
Linearization Technique ◽  
Linear Programming Relaxations ◽  
The One ◽  
Classical Linearization

Abstract A linearization technique for binary quadratic programs (BQPs) that comprise linear constraints is presented. The technique, called “inductive linearization”, extends concepts for BQPs with particular equation constraints, that have been referred to as “compact linearization” before, to the general case. Quadratic terms may occur in the objective function, in the set of constraints, or in both. For several relevant applications, the linear programming relaxations obtained from applying the technique are proven to be at least as strong as the one obtained with a well-known classical linearization. It is also shown how to obtain an inductive linearization automatically. This might be used, e.g., by general-purpose mixed-integer programming solvers.

scholarly journals Data-based polyhedron model for optimization of engineering structures involving uncertainties

2021 ◽  
Vol 2 ◽  
Author(s):  
Zhiping Qiu ◽  
Han Wu ◽  
Isaac Elishakoff ◽  
Dongliang Liu
Keyword(s):  
Linear Programming ◽  
Convex Polyhedron ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Composite Plates ◽  
Convex Polyhedra ◽  
Chebyshev Inequality ◽  
Engineering Structures ◽  
Load Bearing ◽  
Polyhedron Model

Abstract This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

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