On The Reduction of Duality Gap in Box Constrained Nonconvex Quadratic Program

2011 ◽  
Vol 21 (3) ◽  
pp. 706-729 ◽  
Author(s):  
Yong Xia ◽  
Xiaoling Sun ◽  
Duan Li ◽  
Xiaojin Zheng
Keyword(s):  
Duality Gap ◽  
Quadratic Program ◽  
Nonconvex Quadratic Program

scholarly journals Quadratic Program on a Structured Nonconvex Set

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Yi Xu ◽  
Lili Han
Keyword(s):  
Numerical Results ◽  
New Method ◽  
Convex Program ◽  
Quadratic Program ◽  
Small Scale ◽  
Bilevel Program ◽  
Feasible Set ◽  
Low Level ◽  
Nonconvex Quadratic Program ◽  
Upper Level

In this paper, we focus on a special nonconvex quadratic program whose feasible set is a structured nonconvex set. To find an effective method to solve this nonconvex program, we construct a bilevel program, where the low-level program is a convex program while the upper-level program is a small-scale nonconvex program. Utilizing some properties of the bilevel program, we propose a new algorithm to solve this special quadratic program. Finally, numerical results show that our new method is effective and efficient.

scholarly journals A range division and contraction approach for nonconvex quadratic program with quadratic constraints

SpringerPlus ◽  
2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Chunshan Xue ◽  
Hongwei Jiao ◽  
Jingben Yin ◽  
Yongqiang Chen
Keyword(s):  
Quadratic Program ◽  
Quadratic Constraints ◽  
Nonconvex Quadratic Program

scholarly journals On monotonicity and search strategies in face-based copositivity detection algorithms

2021 ◽  
Author(s):  
B. G.-Tóth ◽  
E. M. T. Hendrix ◽  
L. G. Casado
Keyword(s):  
Research Question ◽  
Quadratic Function ◽  
Mixed Integer ◽  
Quadratic Program ◽  
Matrix Methods ◽  
Detection Algorithms ◽  
Standard Simplex ◽  
Mathematical Properties ◽  
The Face ◽  
Spatial Branch And Bound

AbstractOver the last decades, algorithms have been developed for checking copositivity of a matrix. Methods are based on several principles, such as spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the relative interior minima of the standard quadratic program (StQP) and monotonicity. We derive several theoretical properties related to convexity and monotonicity of the standard quadratic function over faces of the standard simplex. We illustrate with numerical instances up to 28 dimensions the use of monotonicity in face-based algorithms. The question is what traversal through the face graph of the standard simplex is more appropriate for which matrix instance; top down or bottom up approaches. This depends on the level of the face graph where the minimum of StQP can be found, which is related to the density of the so-called convexity graph.

Optimization studies of low-density polyethylene process: effect of different interval numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ashraf Azmi ◽  
Suhairi Abdul Sata ◽  
Fakhrony Sholahudin Rohman ◽  
Norashid Aziz
Keyword(s):  
Dynamic Optimization ◽  
Monomer Conversion ◽  
Polymerization Process ◽  
Low Density Polyethylene ◽  
Quadratic Program ◽  
Flow Index ◽  
Low Density ◽  
Orthogonal Collocation ◽  
Interval Numbers ◽  
Optimization Study

AbstractThe highly exothermic nature of the low-density polyethylene (LDPE) polymerization process and the heating-cooling prerequisite in tubular reactor can lead to various problems particularly safety and economic. These issues complicate the monomer conversion maximization approaches. Consequently, the dynamic optimization study to obtain maximum conversion of the LDPE is carried out. A mathematical model has been developed and validated using industrial data. In the dynamic optimization study, maximum monomer conversion (XM) is considered as the objective function, whereas the constraint and bound consists of maximum reaction temperature and product melt flow index (MFI). The orthogonal collocation (OC) on finite elements is used to convert the original optimization problems into Nonlinear Programming (NLP) problems, which are then solved using sequential quadratic program (SQP) methods. The result shows that five interval numbers produce better optimization result compared to one and two intervals.

scholarly journals Decentralized Voltage Optimization Based on the Auxiliary Problem Principle in Distribution Networks with DERs

2021 ◽  
Vol 11 (10) ◽  
pp. 4509
Author(s):  
Anna Rita Di Fazio ◽  
Chiara Risi ◽  
Mario Russo ◽  
Michele De Santis
Keyword(s):  
Distribution Systems ◽  
Auxiliary Problem ◽  
Optimization Procedure ◽  
Distribution Networks ◽  
Small Dimension ◽  
Quadratic Program ◽  
Voltage Profile ◽  
Distributed Energy ◽  
Auxiliary Problem Principle ◽  
Coordination Process

This paper addresses the problem of optimizing the voltage profile of radially-operated distribution systems by acting on the active and reactive powers provided by distributed energy resources (DERs). A novel voltage optimization procedure is proposed by adopting a decentralized control strategy. To this aim, a centralized voltage optimization problem (VOP), minimizing the distance of all the nodal voltages from their reference values, is firstly formulated as a strictly-convex quadratic program. Then, the centralized VOP is rewritten by partitioning the network into voltage control zones (VCZs) with pilot nodes. To overcome the lack of strictly convexity determined by the reduction to the pilot nodes, the dual centralized VOP working on the augmented Lagrangian function is reformulated and iteratively solved by the method of multipliers. Finally, a fully-distributed VOP solution is obtained by applying a distributed algorithm based on the auxiliary problem principle, which allows for solving in each VCZ a quadratic programming problem of small dimension and to drive the VCZ solutions toward the overall optimum by an iterative coordination process that requires to exchange among the VCZs only scalar values. The effectiveness and feasibility of the proposed method have been demonstrated via numerical tests on the IEEE 123-bus system.

Zero duality gap conditions via abstract convexity∗

Optimization ◽  
2021 ◽  
pp. 1-37
Author(s):  
Hoa T. Bui ◽  
Regina S. Burachik ◽  
Alexander Y. Kruger ◽  
David T. Yost
Keyword(s):  
Duality Gap ◽  
Abstract Convexity ◽  
Zero Duality Gap

A bound on the roof-duality gap

1989 ◽  
pp. 254-257 ◽  
Author(s):  
P. L. Hammer ◽  
B. Kalantari
Keyword(s):  
Duality Gap

scholarly journals An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

2021 ◽  
Author(s):  
E. Alper Yıldırım
Keyword(s):  
Objective Function ◽  
Convex Relaxation ◽  
Large Family ◽  
Quadratic Program ◽  
Feasible Region ◽  
Recession Cone ◽  
Convex Relaxations ◽  
Quadratic Programs ◽  
Alternative Perspective ◽  
Optimal Value

AbstractWe study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

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