scholarly journals Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints

2005 ◽  
Vol 1 (1) ◽  
pp. 53-63 ◽  
Author(s):  
David Yang Gao ◽  
Keyword(s):  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Nonconvex Minimization ◽  
Perfect Duality

A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

2008 ◽  
Vol 24 (3) ◽  
pp. 823-828 ◽  
Author(s):  
Henghsiu Tsai ◽  
Kung-Sik Chan
Keyword(s):  
Generating Function ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Conditional Variance ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Economic Statistics ◽  
Absolute Monotonicity ◽  
Necessary And Sufficient ◽  
The Garch Model

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH(p,q) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH(p,q) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH(p,q) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

scholarly journals Limited-Budget Formation Control for High-Order Linear Swarm Systems with Fix Topologies

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Hongtao Dang ◽  
Le Wang ◽  
Yan Zhang ◽  
Jianye Yang
Keyword(s):  
Formation Control ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
High Order ◽  
Linear Matrix ◽  
Matrix Inequality ◽  
Order Linear ◽  
Limited Budget ◽  
Communication Topology ◽  
Formation Design

This paper discusses limited-budget time-varying formation design and analysis problems for a high-order linear swarm system with a fixed communication topology. Firstly, the communication topology among agents is modeled as an undirected and connected graph, and a new formation control protocol with an energy integral term is proposed to realize formation control and to guarantee the practical energy assumption is less than the limited energy budget. Then, by the matrix inequality tool, sufficient conditions for limited-budget formation design and analysis are proposed, respectively, which are scalable and checkable since they are independent of the number of agents of a swarm system and can be transformed into linear matrix inequality constraints. Moreover, an explicit expression of the formation center function is given, which contains the formation function part and the cooperative state part and is not associated with the derivatives of the formation functions. Finally, a numerical simulation is shown to demonstrate the effectiveness of theoretical results.

scholarly journals On G-invexity-type nonlinear programming problems

2015 ◽  
Vol 5 (1) ◽  
pp. 13-20
Author(s):  
Tadeusz Antczak ◽  
Manuel Arana Jiménez
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Extremum Problem ◽  
Sufficient Optimality Conditions ◽  
New Class ◽  
Necessary And Sufficient ◽  
Wolfe Dual

In this paper, we introduce the concepts of KT-G-invexity and WD$-G-invexity for the considered differentiable optimization problem with inequality constraints. Using KT-G-invexity notion, we prove new necessary and sufficient optimality conditions for a new class of such nonconvex differentiable optimization problems. Further, the so-called G-Wolfe dual problem is defined for the considered extremum problem with inequality constraints. Under WD-G-invexity assumption, the necessary and sufficient conditions for weak duality between the primal optimization problem and its G-Wolfe dual problem are also established.

scholarly journals Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives

2021 ◽  
Author(s):  
Tung Nguyen
Keyword(s):  
Optimization Problems ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Second Order ◽  
Set Valued Mapping ◽  
Contingent Epiderivative ◽  
Wolfe Duality ◽  
Order Constraint ◽  
Sufficient Conditions For Optimality ◽  
Second Order Necessary Conditions

We propose a generalized second-order asymptotic contingent epiderivative of a set-valued mapping, study its properties, as well as relations to some second-order contingent epiderivatives, and sufficient conditions for its existence. Then, using these epiderivatives, we investigate set-valued optimization problems with generalized inequality constraints. Both second-order necessary conditions and sufficient  conditions for optimality of the Karush-Kuhn-Tucker type are established under the second-order constraint qualification. An application to Mond-Weir and Wolfe duality schemes is also presented. Some remarks and examples are provided to illustrate our results.

Optimal Control of Linear Distributed Parameter Systems With Constrained Inputs

1969 ◽  
Vol 91 (2) ◽  
pp. 161-167 ◽  
Author(s):  
W. A. Weigand ◽  
A. F. D’Souza
Keyword(s):  
Optimal Control ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Inequality Constraints ◽  
Open Loop ◽  
Fredholm Integral Equations ◽  
Loop Control ◽  
Linear Partial Differential Equations ◽  
Control Functions ◽  
Unconstrained Problem

The optimal open loop control of systems described by a set of linear partial differential equations is investigated. The performance index is of quadratic type and the mean square error is considered as a special case. Energy type inequality constraints are imposed on the control inputs. The problem is formulated as a minimization problem in Hilbert space. The necessary and sufficient conditions for a minimum are obtained and it is proved that these conditions yield the global minimum. It is shown how the solution to the constrained problem can be obtained from the solution of the unconstrained problem. The optimal control functions satisfy Fredholm integral equations with symmetric kernels. The paper presents an example where the solution is obtained by eigenfunction expansion.

Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

2021 ◽  
pp. 1-16
Author(s):  
Sunae Pak ◽  
Huichol Choe ◽  
Kinam Sin ◽  
Sunghyok Kwon
Keyword(s):  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Initial Value ◽  
Solution Representation ◽  
Necessary And Sufficient

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

Recovery of Sparse Signal and Nonconvex Minimization

2014 ◽  
Vol 651-653 ◽  
pp. 2177-2180
Author(s):  
Jia Jing ◽  
Jian Jun Wang
Keyword(s):  
Sufficient Conditions ◽  
Restricted Isometry Property ◽  
Sparse Signal ◽  
Nonconvex Minimization ◽  
Perfect Recovery

In this paper, we present sufficient conditions in term of the restricted isometry property (RIP) to guarantee perfect recovery of sparse signal in the noiseless case and stable recovery in the noisy case via Lq-minimization, especially for nonconvex case 0<q<1. Using RIP condition, we present sufficient conditions to guarantee perfect recovery of sparse signal in the noiseless case and stable recovery in the noisy case via Lq-minimization.

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