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.

scholarly journals On standard quadratic programs with exact and inexact doubly nonnegative relaxations

2021 ◽  
Author(s):  
Y. Görkem Gökmen ◽  
E. Alper Yıldırım
Keyword(s):  
Positive Semidefinite ◽  
Quadratic Program ◽  
Algebraic Characterization ◽  
Nonnegative Matrices ◽  
Quadratic Programs ◽  
Optimal Value ◽  
Positive Matrices ◽  
Doubly Nonnegative Relaxation ◽  
Standard Quadratic Program

AbstractThe problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over the computationally intractable cone of completely positive matrices. Replacing the intractable cone in this formulation by the larger but tractable cone of doubly nonnegative matrices, i.e., the cone of positive semidefinite and componentwise nonnegative matrices, one obtains the so-called doubly nonnegative relaxation, whose optimal value yields a lower bound on that of the original problem. We present a full algebraic characterization of the set of instances of standard quadratic programs that admit an exact doubly nonnegative relaxation. This characterization yields an algorithmic recipe for constructing such an instance. In addition, we explicitly identify three families of instances for which the doubly nonnegative relaxation is exact. We establish several relations between the so-called convexity graph of an instance and the tightness of the doubly nonnegative relaxation. We also provide an algebraic characterization of the set of instances for which the doubly nonnegative relaxation has a positive gap and show how to construct such an instance using this characterization.

scholarly journals Application of Change of Basis in the Simplex Method

2017 ◽  
Vol 11 (1) ◽  
pp. 41
Author(s):  
Mehmet Hakan Özdemir
Keyword(s):  
Linear Programming ◽  
Iterative Method ◽  
Objective Function ◽  
Simplex Method ◽  
Programming Model ◽  
Feasible Region ◽  
Optimal Value ◽  
Linear Programming Problems ◽  
Repeated Use ◽  
Simplex Tableau

The simplex method is a very useful method to solve linear programming problems. It gives us a systematic way of examining the vertices of the feasible region to determine the optimal value of the objective function. It is executed by performing elementary row operations on a matrix that we call the simplex tableau. It is an iterative method that by repeated use gives us the solution to any n variable linear programming model. In this paper, we apply the change of basis to construct following simplex tableaus without applying elementary row operations on the initial simplex tableau.

scholarly journals Research on the inverse kinematics of manipulator using an improved self-adaptive mutation differential evolution algorithm

2021 ◽  
Vol 18 (3) ◽  
pp. 172988142110144
Author(s):  
Qianqian Zhang ◽  
Daqing Wang ◽  
Lifu Gao
Keyword(s):  
Differential Evolution ◽  
Objective Function ◽  
Inverse Kinematics ◽  
Differential Evolution Algorithm ◽  
Weight Coefficient ◽  
Adaptive Mutation ◽  
Feasible Region ◽  
Serial Manipulators ◽  
De Algorithm ◽  
Self Adaptive

To assess the inverse kinematics (IK) of multiple degree-of-freedom (DOF) serial manipulators, this article proposes a method for solving the IK of manipulators using an improved self-adaptive mutation differential evolution (DE) algorithm. First, based on the self-adaptive DE algorithm, a new adaptive mutation operator and adaptive scaling factor are proposed to change the control parameters and differential strategy of the DE algorithm. Then, an error-related weight coefficient of the objective function is proposed to balance the weight of the position error and orientation error in the objective function. Finally, the proposed method is verified by the benchmark function, the 6-DOF and 7-DOF serial manipulator model. Experimental results show that the improvement of the algorithm and improved objective function can significantly improve the accuracy of the IK. For the specified points and random points in the feasible region, the proportion of accuracy meeting the specified requirements is increased by 22.5% and 28.7%, respectively.

A new solution method for a class of large dimension rank-two nonconvex programs

2020 ◽  
Author(s):  
Riccardo Cambini ◽  
Irene Venturi
Keyword(s):  
Objective Function ◽  
Optimal Level ◽  
Outer Approximation ◽  
Point Of View ◽  
Large Dimension ◽  
Low Rank ◽  
Feasible Region ◽  
Quantitative Management ◽  
Solution Methods ◽  
Optimal Level Solutions

Abstract Low-rank problems are nonlinear minimization problems in which the objective function, by means of a suitable linear transformation of the variables, depends on very few variables. These problems often arise in quantitative management science applications, for example, in location models, transportation problems, production planning, data envelopment analysis and multiobjective programs. They are usually approached by means of outer approximation, branch and bound, branch and select and optimal level solution methods. The paper studies, from both a theoretical and an algorithmic point of view, a class of large-dimension rank-two nonconvex problems having a polyhedral feasible region and $f(x)=\phi (c^Tx+c_0,d^Tx+d_0)$ as the objective function. The proposed solution algorithm unifies a new partitioning method, an outer approximation approach and a mixed method. The results of a computational test are provided to compare these three approaches with the optimal level solutions method. In particular, the new partitioning method performs very well in solving large problems.

New Approach for the Continuing Dynamic Programming

2012 ◽  
Vol 459 ◽  
pp. 575-578
Author(s):  
Peng Zhang ◽  
Xiang Huan Meng
Keyword(s):  
Dynamic Programming ◽  
Objective Function ◽  
Iteration Method ◽  
Programming Model ◽  
State Equations ◽  
New Approach ◽  
Optimal Value ◽  
Convergent Method ◽  
Optimal Values ◽  
Dynamic Programming Model

The paper proposes the discrete approximate iteration method to solve single-dimensional continuing dynamic programming model. The paper also presents a comparison of the discrete approximate iteration method and bi- convergent method to solve multi-dimensional continuing dynamic programming model. The algorithm is the following: Firstly, let state value of one of state equations be unknown and the others be known. Secondly, use discrete approximate iteration method to find the optimal value of the unknown state values, continue iterating until all state equations have found optimal values. If the objective function is convex, the algorithm is proved linear convergent. If the objective function is non-concave and non-convex, the algorithm is proved convergent.

Computer Control to Manage Risk of Pedestrian Head Crash Management Systems Based-High Risky Spots

2013 ◽  
Vol 321-324 ◽  
pp. 1982-1985
Author(s):  
Hong Ying Ji
Keyword(s):  
Objective Function ◽  
Shortest Path ◽  
Contact Area ◽  
Computer Control ◽  
Management Systems ◽  
Feasible Region ◽  
Forming Simulation ◽  
Drawing Direction ◽  
Optimal Direction ◽  
The Right

The right panel drawing direction is an important prerequisite for generating qualified parts, an important step before the panel forming simulation is to determine the reasonable direction of the drawing. Manually adjust parts in order to overcome rely on experience, the drawbacks to the drawing direction, the direction of the drawing punch and forming the contact area of the sheet as the goal of Computer Control determination Shortest Path Method. Objective function of the direction of the drawing for the variable contact area in the drawing direction of the feasible region, the use of heritage Shortest Path Methods to optimize the objective function of the contact area and, ultimately feasible within the contact area corresponding to the drawing direction, that is the best drawing direction. The measured results show that the direction of the drawing based on genetic Shortest Path Method, the Computer Control Shortest Path Method can fast and accurate to obtain the optimal direction of drawing.

A Generalized Return Vector for Projection Methods in Optimization With Nonlinear Constraints

1976 ◽  
Vol 98 (3) ◽  
pp. 816-819
Author(s):  
K. T. A. Ho ◽  
M. A. Townsend
Keyword(s):  
Objective Function ◽  
Nonlinear Optimization ◽  
Projection Methods ◽  
Feasible Region ◽  
Nonlinear Constraints ◽  
Variable Metric Methods ◽  
Variable Metric ◽  
Numerical Example ◽  
Constrained Nonlinear Optimization

Variable metric methods can be adapted to constrained nonlinear optimization by incorporating projection methods and a return vector when the indicated next step leaves the feasible region. A generalized return vector is developed here which yields a superior return to the feasible region in terms of the metric associated with the objective function. It is shown that a better point results and faster convergence is expected. A numerical example is given.

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