scholarly journals Branch-and-Reduction Algorithm for Indefinite Quadratic Programming Problem

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Yongjian Qiu ◽  
Yuming Zhu ◽  
Jingben Yin
Keyword(s):  
Quadratic Programming ◽  
Programming Problem ◽  
Optimal Solution ◽  
Quadratic Programming Problem ◽  
Reduction Algorithm ◽  
Indefinite Quadratic Programming ◽  
Computational Performance ◽  
Indefinite Quadratic Programming Problem ◽  
Global Optimal ◽  
Indefinite Quadratic

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.

scholarly journals On the closure of the sum of closed subspaces

2001 ◽  
Vol 26 (5) ◽  
pp. 257-267 ◽  
Author(s):  
Irwin E. Schochetman ◽  
Robert L. Smith ◽  
Sze-Kai Tsui
Keyword(s):  
Hilbert Space ◽  
Quadratic Programming ◽  
Programming Problem ◽  
Sufficient Conditions ◽  
Optimal Solution ◽  
The Other ◽  
Quadratic Programming Problem ◽  
Orthogonal Projections ◽  
The Cost ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into the orthogonal complement of the other is closed. We also give sufficient conditions for the sum to be closed in terms of the relevant orthogonal projections. As a consequence, we obtain sufficient conditions for the existence of an optimal solution to an abstract quadratic programming problem in terms of the kernels of the cost and constraint operators.

scholarly journals Unconstrained Quadratic Programming Problem with Uncertain Parameters

2021 ◽  
pp. 1-7
Author(s):  
Sie Long Kek ◽  
Fong Peng Lim ◽  
Harley Ooi
Keyword(s):  
Quadratic Programming ◽  
Programming Problem ◽  
Linear Equations ◽  
Optimal Solution ◽  
Theoretical Work ◽  
Quadratic Programming Problem ◽  
Model Parameters ◽  
Optimization Approach ◽  
Uncertain Parameters ◽  
Unconstrained Quadratic Programming

In this paper, an unconstrained quadratic programming problem with uncertain parameters is discussed. For this purpose, the basic idea of optimizing the unconstrained quadratic programming problem is introduced. The solution method of solving linear equations could be applied to obtain the optimal solution for this kind of problem. Later, the theoretical work on the optimization of the unconstrained quadratic programming problem is presented. By this, the model parameters, which are unknown values, are considered. In this uncertain situation, it is assumed that these parameters are normally distributed; then, the simulation on these uncertain parameters are performed, so the quadratic programming problem without constraints could be solved iteratively by using the gradient-based optimization approach. For illustration, an example of this problem is studied. The computation procedure is expressed, and the result obtained shows the optimal solution in the uncertain environment. In conclusion, the unconstrained quadratic programming problem, which has uncertain parameters, could be solved successfully.

Quadratic hyper-surface kernel-free least squares support vector regression

2021 ◽  
Vol 25 (2) ◽  
pp. 265-281
Author(s):  
Junyou Ye ◽  
Zhixia Yang ◽  
Zhilin Li
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Support Vector Regression ◽  
Programming Problem ◽  
Linear Equations ◽  
Optimal Solution ◽  
Quadratic Function ◽  
Regression Function ◽  
Quadratic Programming Problem ◽  
Support Vector

We present a novel kernel-free regressor, called quadratic hyper-surface kernel-free least squares support vector regression (QLSSVR), for some regression problems. The task of this approach is to find a quadratic function as the regression function, which is obtained by solving a quadratic programming problem with the equality constraints. Basically, the new model just needs to solve a system of linear equations to achieve the optimal solution instead of solving a quadratic programming problem. Therefore, compared with the standard support vector regression, our approach is much efficient due to kernel-free and solving a set of linear equations. Numerical results illustrate that our approach has better performance than other existing regression approaches in terms of regression criterion and CPU time.

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