Sufficient conditions for global optimality of bivalent nonconvex quadratic programs with inequality constraints
- Authors: Wu, Zhiyou , Jeyakumar, Vaithilingam , Rubinov, Alex
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 133, no. 1 (2007), p. 123-130
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- Description: We present sufficient conditions for the global optimality of bivalent nonconvex quadratic programs involving quadratic inequality constraints as well as equality constraints. By employing the Lagrangian function, we extend the global subdifferential approach, developed recently in Jeyakumar et al. (J. Glob. Optim., 2007, to appear; Math. Program. Ser. A, 2007, to appear) for studying bivalent quadratic programs without quadratic constraints, and derive global optimality conditions. © 2007 Springer Science+Business Media, LLC.
- Description: C1
A filled function method for nonlinear equations
- Authors: Wu, Zhiyou , Mammadov, Musa , Bai, Fusheng , Yang, Y. J.
- Date: 2007
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 189, no. 2 (2007), p. 1196-1204
- Full Text: false
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- Description: In this paper, we propose a new global optimization approach based on the filled function method for solving box-constrained systems of nonlinear equations. The special properties of optimization problem are employed to construct a novel filled function. The objective function value can be reduced by half in each iteration of our filled function algorithm. Several numerical examples are presented to illustrate the efficiency of the present approach.
- Description: C1
- Description: 2003005618
A filled function method for constrained global optimization
- Authors: Wu, Zhiyou , Bai, Fusheng , Lee, Heung , Yang, Yongjian
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 39, no. 4 (2007), p. 495-507
- Full Text: false
- Reviewed:
- Description: In this paper, a filled function method for solving constrained global optimization problems is proposed. A filled function is proposed for escaping the current local minimizer of a constrained global optimization problem by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a filled function method for obtaining a global minimizer or an approximate global minimizer of the constrained global optimization problem is presented. Some numerical results demonstrate the efficiency of this global optimization method for solving constrained global optimization problems. © 2007 Springer Science+Business Media, Inc.
- Description: C1
- Description: 2003005513
Optimality conditions in global optimization and their applications
- Authors: Rubinov, Alex , Wu, Zhiyou
- Date: 2009
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 120, no. 1 SPEC. ISS. (2009), p. 101-123
- Full Text: false
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- Description: In this paper we derive necessary and sufficient conditions for some problems of global minimization. Our approach is based on methods of abstract convexity: we use a representation of an upper semicontinuous function as the lower envelope of a family of convex functions. We discuss applications of conditions obtained to the examination of some tractable sufficient conditions for the global minimum and to the theory of inequalities. © 2007 Springer-Verlag.
A filled function method for constrained nonlinear integer programming
- Authors: Yang, Yongjian , Wu, Zhiyou , Bai, Fusheng
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Industrial and Management Optimization Vol. 4, no. 2 (May 2008), p. 353-362
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- Description: A filled function method is presented in this paper to solve constrained nonlinear integer programming problems. It is shown that for a given non-global local minimizer, a better local minimizer can be obtained by local search staring from an improved initial point which is obtained by locally solving a box-constrained integer programming problem. Several illustrative numerical examples are reported to show the efficiency of the present method.
- Description: C1
Global optimality conditions for mixed integer weakly concave programming problems
- Authors: Wu, Zhiyou , Quan, Jing , Bai, Fusheng
- Date: 2010
- Type: Text , Journal article
- Relation: Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 17, no. 5 (2010), p. 675-685
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- Description: In this paper, some necessary and some suffcient global optimality conditions for a class of mixed integer programming problems whose objective functions are the difference of quadratic functions and convex functions are established. The numerical examples are also presented to show the significance of the global optimality conditions for this class of programming problems. Copyright © 2010 Watam Press.
A filled function method for quadratic programs with binary constraints
- Authors: Wu, Zhiyou , Yang, Y. J. , Bai, Fusheng
- Date: 2009
- Type: Text , Journal article
- Relation: Optimization Vol. 58, no. 5 (2009), p. 585-598
- Full Text: false
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- Description: In this article, we first introduce a definition of a local minimizer for the quadratic programs with {-1, 1} constraints and show that the existing necessary condition of global optimality in some recent literature is in fact the necessary and sufficient condition of local optimality under the present definition. A filled function and the corresponding filled function algorithm then are proposed to get the global minimizer of the concerned minimization problem. The existing sufficient condition of global optimality is employed to form one of the termination rules of the algorithm. Several numerical examples are reported to show the efficiency of the present method for solving quadratic programs with binary constraints.
A class of convexification and concavification methods for non-monotone optimization problems
- Authors: Wu, Zhiyou , Lee, Heung , Yang, Xin-Min
- Date: 2005
- Type: Text , Journal article
- Relation: Optimization Vol. 54, no. 6 (2005), p. 605-625
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- Description: A class of convexification and concavification methods are proposed for solving some classes of non-monotone optimization problems. It is shown that some classes of non-monotone optimization problems can be converted into better structured optimization problems, such as, concave minimization problems, reverse convex programming problems, and canonical D.C. programming problems by the proposed convexification and concavification methods. The equivalence between the original problem and the converted better structured optimization problem is established.
- Description: C1
- Description: 2003003608
A novel monotonization transformation for some classes of global optimization problems
- Authors: Bai, Fusheng , Wu, Zhiyou
- Date: 2006
- Type: Text , Journal article
- Relation: Asia-Pacific Journal of Operational Research Vol. 23, no. 3 (Sep 2006), p. 371-392
- Full Text: false
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- Description: A novel monotonization method is proposed for converting a non-monotone programming problem into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavification methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.
- Description: C1
- Description: 2003003590
Global optimality conditions for some classes of optimization problems
- Authors: Wu, Zhiyou , Rubinov, Alex
- Date: 2009
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 145, no. 1 (2009), p. 164-185
- Full Text: false
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- Description: We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity. © 2009 Springer Science+Business Media, LLC.
Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions
- Authors: Jeyakumar, Vaithilingam , Rubinov, Alex , Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 110, no. 3 (2007), p. 521-541
- Full Text: false
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- Description: In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. © Springer-Verlag 2007.
- Description: C1
A new local and global optimization method for mixed integer quadratic programming problems
- Authors: Li, G. Q. , Wu, Zhiyou , Quan, Jing
- Date: 2010
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 217, no. 6 (2010), p. 2501-2512
- Full Text: false
- Reviewed:
- Description: In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable. Crown Copyright © 2010.
A dual criterion for maximal monotonicity of composition operators
- Authors: Jeyakumar, Vaithilingam , Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Set-Valued Analysis Vol. 15, no. 3 (2007), p. 265-273
- Full Text: false
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- Description: In this paper we present a dual criterion for the maximal monotonicity of the composition operator T:=A* SA, where S:Y→→ Y is a maximal monotone (set-valued) operator and A: X→ Y is a continuous linear map with the adjoint A*, X and Y are reflexive Banach spaces, and the product notation indicates composition. The dual criterion is expressed in terms of the closure condition involving the epigraph of the conjugate of Fitzpatrick function associated with S, and the operator A. As an easy application, a dual criterion for the maximality of the sum of two maximal monotone operators is also given. © 2006 Springer Science+Business Media B.V.
- Description: C1
Global optimality conditions for mixed nonconvex quadratic programs
- Authors: Wu, Zhiyou , Bai, Fusheng
- Date: 2009
- Type: Text , Journal article
- Relation: Optimization Vol. 58, no. 1 (2009), p. 39-47
- Full Text: false
- Reviewed:
- Description: In this article, we present some global optimality conditions for mixed quadratic programming problems. Our approach is based on a L-subdifferential and an associated L-normal cone. Unlike most subdifferentials, the L-subdifferential is formed by functions that are not necessarily linear functions. We derive some sufficient and necessary global optimality conditions for mixed quadratic programs with box constraints and binary constraints.
Necessary and sufficient conditions for stable conjugate duality
- Authors: Burachik, Regina , Jeyakumar, Vaithilingam , Wu, Zhiyou
- Date: 2006
- Type: Text , Journal article
- Relation: Journal of Nonlinear Analysis Vol. 64, no. 9 (2006), p. 1998-2005
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- Description: The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ' −φ∗(0,v), whenever a regularity condition on φ is satisfied, is a key result in convex anal¬ysis and optimization, where φ : X × Y → IR ∪{+∞} is a convex function, X and Y are Banach spaces, Y ' is the continuous dual space of Y and φ∗ is the Fenchel-Moreau conjugate of φ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, ∗ ∗ ∈ X' inf {φ(x, 0) + x ∗(x)} = max {−φ ∗(−x ,v)}, ∀x, x∈Xv∈Y ' and obtain a new global dual regularity condition, which is much more general than the popularly known interior-point type conditions, for the conjugate duality. As a consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel-Rockafellar duality theorem. In the case where one of the functions involved in the duality is a polyhedral convex function, we also provide generalized interior-point conditions for the epigraph closure condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. As applications, we give general sufficient conditions for a minimax theorem, a subdifferential composition formula and for duality results of convex programming problems.
- Description: C1
- Description: 2003003596
Generalized Fenchel's conjugation formulas and duality for abstract convex functions
- Authors: Jeyakumar, Vaithilingam , Rubinov, Alex , Wu, Zhiyou
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 132, no. 3 (Mar 2007), p. 441-458
- Full Text: false
- Reviewed:
- Description: In this paper, we present a generalization of Fenchel's conjugation and derive infimal convolution formulas, duality and subdifferential (and epsilon-subdifferential) sum formulas for abstract convex functions. The class of abstract convex functions covers very broad classes of nonconvex functions. A nonaffine global support function technique and an extended sum- epiconjugate technique of convex functions play a crucial role in deriving the results for abstract convex functions. An additivity condition involving global support sets serves as a constraint qualification for the duality.
- Description: C1
A filled function method for constrained nonlinear equations
- Authors: Bai, Fusheng , Mammadov, Musa , Wu, Zhiyou , Yang, Yongjian
- Date: 2008
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 4, no. 1 (Jan 2008), p. 9-18
- Full Text: false
- Reviewed:
- Description: We consider the problem of solving a constrained system of nonlinear equations. After reformulating the system into an equivalent constrained global optimization problems, we construct a filled function based on a special property of the reformulated problem. A filled function method is then proposed to solve the constrained system of nonlinear equations. Some numerical examples are presented to illustrate the usefulness of the present techniques.
- Description: C1
Lower order calmness and exact penalty function
- Authors: Bai, Fusheng , Wu, Zhiyou , Zhu, D.
- Date: 2006
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 21, no. 4 (2006), p. 515-526
- Full Text: false
- Reviewed:
- Description: In this article, we investigate the exact penalty properties of a lower order penalty function under a lower order calmness conditions. It is shown that the local exact penalization of the lower order penalty function with any positive penalty parameter holds under the local lower order calmness condition. A necessary and sufficient condition for global exact penalization of the lower order penalty function is given in terms of global lower order calmness condition. Furthermore, a formula of least global exact penalty parameter for the lower order penalty function is obtained.
- Description: C1
- Description: 2003002855
Probabilistic neural networks based network security management
- Authors: Wu, Zhiyou , Liu, W , Wu, Jian-ping , Duan, Hai-xin
- Date: 2008
- Type: Text , Journal article
- Relation: Journal Communication and Computer Vol. 5, no. 2 (2008), p. 19-24
- Full Text: false
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- Description: Intrusion Detection System (IDS) is one of the main tools in computer and network management, we consider intrusion detection using probabilistic neural networks. An ensemble of Probabilistic Neural Networks (PNN) is trained with Adaptive Boost to classify the detected event as normal or intrusive. We use Hamming distance kernels for PNN and find them superior to Euclidean distance kernels for this kind of detected event
Global descent methods for unconstrained global optimization
- Authors: Wu, Zhiyou , Li, Duan , Zhang, Lian-Sheng
- Date: 2011
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 50, no. 3 (2011), p. 379-3976
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- Description: We propose in this paper novel global descent methods for unconstrained global optimization problems to attain the global optimality by carrying out a series of local minimization. More specifically, the solution framework consists of a two-phase cycle of local minimization: the first phase implements local search of the original objective function, while the second phase assures a global descent of the original objective function in the steepest descent direction of a (quasi) global descent function. The key element of global descent methods is the construction of the (quasi) global descent functions which possess prominent features in guaranteeing a global descent. © 2010 Springer Science+Business Media, LLC.