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PERFECT DUALITY THEORY AND COMPLETE SOLUTIONS TO A CLASS OF GLOBAL OPTIMIZATION PROBLEMS
, 2003
"... This article presents a complete set of solutions for a class of global optimization problems. These problems are directly related to numericalization of a large class of semilinear nonconvex partial differential equations in nonconvex mechanics including phase transitions, chaotic dynamics, nonline ..."
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Cited by 24 (14 self)
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This article presents a complete set of solutions for a class of global optimization problems. These problems are directly related to numericalization of a large class of semilinear nonconvex partial differential equations in nonconvex mechanics including phase transitions, chaotic dynamics, nonlinear field theory, and superconductivity. The method used is the socalled canonical dual transformation developed recently. It is shown that, by this method, these difficult nonconvex constrained primal problems in R n can be converted into a onedimensional canonical dual problem, i.e. the perfect dual formulation with zero duality gap and without any perturbation. This dual criticality condition leads to an algebraic equation which can be solved completely. Therefore, a complete set of solutions to the primal problems is obtained. The extremality of these solutions are controlled by the triality theory discovered recently [D.Y. Gao (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications, Vol. xviii, p. 454. Kluwer Academic Publishers, Dordrecht/Boston/London.]. Several examples are illustrated including the nonconvex constrained quadratic programming. Results show that these problems can be solved completely to obtain all KKT points and global minimizers.
NONCONVEX SEMILINEAR PROBLEMS AND CANONICAL DUALITY SOLUTIONS
"... This paper presents a brief review and some new developments on the canonical duality theory with applications to a class of variational problems in nonconvex mechanics and global optimization. These nonconvex problems are directly related to a large class of semilinear partial differential equatio ..."
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Cited by 19 (16 self)
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This paper presents a brief review and some new developments on the canonical duality theory with applications to a class of variational problems in nonconvex mechanics and global optimization. These nonconvex problems are directly related to a large class of semilinear partial differential equations in mathematical physics including phase transitions, postbuckling of large deformed beam model, chaotic dynamics, nonlinear field theory, and superconductivity. Numerical discretizations of these equations lead to a class of very difficult global minimization problems in finite dimensional space. It is shown that by the use of the canonical dual transformation, these nonconvex constrained primal problems can be converted into certain very simple canonical dual problems. The criticality condition leads to dual algebraic equations which can be solved completely. Therefore, a complete set of solutions to these very difficult primal problems can be obtained. The extremality of these solutions are controlled by the socalled triality theory. Several examples are illustrated including the nonconvex constrained quadratic programming. Results show that these very difficult primal problems can be converted into certain simple canonical (either convex or concave) dual problems, which can be solved completely. Also some very interesting new phenomena, i.e. triochaos and metachaos, are discovered in postbuckling of nonconvex systems. The author believes that these important phenomena exist in many nonconvex dynamical systems and deserve to have a detailed study.
Canonical Duality Theory and Solutions to Constrained Nonconvex Quadratic Programming
, 2004
"... This paper presents a perfect duality theory and a complete set of solutions to nonconvex quadratic programming problems subjected to inequality constraints. By use of the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal ..."
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Cited by 12 (7 self)
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This paper presents a perfect duality theory and a complete set of solutions to nonconvex quadratic programming problems subjected to inequality constraints. By use of the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal problem in the sense that they have the same set of KKT points. It is proved that the KKT points depend on the index of the Hessian matrix of the total cost function. The global and local extrema of the nonconvex quadratic function can be identified by the triality theory [11]. Results show that if the global extrema of the nonconvex quadratic function are located on the boundary of the primal feasible space, the dual solutions should be interior points of the dual feasible set, which can be solved by deterministic methods. Certain nonconvex quadratic programming problems in � n can be converted into a dual problem with only one variable. It turns out that a complete set of solutions for quadratic programming over a sphere is obtained as a byproduct. Several examples are illustrated.
On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian
 Journal of Global Optimization
, 2006
"... We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condit ..."
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Cited by 8 (2 self)
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We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the stepsize parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. Key words: Nonconvex programming; nonsmooth optimization; augmented Lagrangian; sharp Lagrangian; subgradient optimization.
c ○ TÜB˙ITAK Solving Fuzzy Linear Programming Problems with Linear Membership Functions
"... In this paper, we concentrate on two kinds of fuzzy linear programming problems: linear programming problems with only fuzzy technological coefficients and linear programming problems in which both the righthand side and the technological coefficients are fuzzy numbers. We consider here only the ca ..."
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Cited by 8 (0 self)
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In this paper, we concentrate on two kinds of fuzzy linear programming problems: linear programming problems with only fuzzy technological coefficients and linear programming problems in which both the righthand side and the technological coefficients are fuzzy numbers. We consider here only the case of fuzzy numbers with linear membership functions. The symmetric method of Bellman and Zadeh [2] is used for a defuzzification of these problems. The crisp problems obtained after the defuzzification are nonlinear and even nonconvex in general. We propose here the “modified subgradient method ” and use it for solving these problems. We also compare the new proposed method with well known “fuzzy decisive set method”. Finally, we give illustrative examples and their numerical solutions. Key Words: Fuzzy linear programming; fuzzy number; modified subgradient method; fuzzy decisive set method.
An Inexact Modified Subgradient Algorithm for Nonconvex Optimization ∗
, 2008
"... We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence ..."
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Cited by 4 (1 self)
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We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may allow to solve problems with less computational effort. We illustrate this through test problems, including an optimal bang–bang control problem, under several different inexactness schemes.
Augmented Lagrangian Coordination for Decomposed Design Problems
 6TH WORLD CONGRESSES OF STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION, RIO DE JANEIRO. PAPER
, 2005
"... Designing largescale systems frequently involves solving a complex mathematical program that requires, for var iousr easons, decomposition into a number of smaller systems. Practical studies have proved the effectiveness of multilevel hierarchical methods at early stages of design; these methods di ..."
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Cited by 3 (0 self)
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Designing largescale systems frequently involves solving a complex mathematical program that requires, for var iousr easons, decomposition into a number of smaller systems. Practical studies have proved the effectiveness of multilevel hierarchical methods at early stages of design; these methods divide a large pr ogr am into multiple levels and multiple systems at each level and ther esult is known as a multilevel or hier ar chical mathematical pr ogr am. Using insight pr ovided by Lagr angian duali ty theor y, the paper pr esents a new algor ithm for bilevel pr ogr ams that does not impose difficulttosatisfy conditions of convexit y and differ entiability gener ally r equir ed for conver gence. In addition, the algor ith mr ender s each subpr oblem independent so that they can be solved concur r ently, which is a significant advantage in cer tain applications. This is done by combining classical Lagrangian duality (LD) and the augmented Lagrangian duality (ALD) in an algor ithm to solve a bilevel problem. One of the tr aditional drawbacks of LD has been that it is only applicable to convex problems. ALD extends the theory to nonconvex pr oblems but at the expense of separability. Combining classical LD with ALD provides a simple method for decomposition without imposing restrictive conditions. The method is applied to two mathematical examples to illustrate its potential as well as associated numerical isues.
Stabilizing Acrobot by Using Nonlinear Programming Based on Sliding Mode Controller
"... Proceedings, pp. 712—724 We design sliding mode controllers for nonlinear dynamic systems by using a nonlinear programming approach. We show that by appropriate selection of the objective function and the constraints, it is possible to obtain sliding mode controller parameters by solving a sequence ..."
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Proceedings, pp. 712—724 We design sliding mode controllers for nonlinear dynamic systems by using a nonlinear programming approach. We show that by appropriate selection of the objective function and the constraints, it is possible to obtain sliding mode controller parameters by solving a sequence of nonlinear programming problems. These parameters determine the forcing function which satisfies possibly nonlinear, even nonconvex constraints and optimize a given nonlinear objective function. We use the Modified Subgradient Algorithm for the nonconvex optimization problems encountered in computing such forcing functions. We illustrate the validity of our approach by stabilizing an underactuated two link robot manipulator, called Acrobot, at vertically upright position.
A Deflected Subgradient Method Using a General Augmented Lagrangian Duality With Implications on Penalty Methods
, 2009
"... We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primaldual method with strong duality, i.e., with z ..."
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We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primaldual method with strong duality, i.e., with zero duality gap. We prove that our method converges to a dual solution if and only if a dual solution exists. We also prove that all accumulation points of an auxiliary primal sequence are primal solutions. Our results apply, in particular, to classical penalty methods, since the penalty functions associated with these methods can be recovered as a special case of our augmented Lagrangians. Besides the classical augmenting terms given by the 1 or 2norm forms, terms of many other forms can be used in our Lagrangian function. Using a practical selection of the stepsize parameters, as well as various choices of the augmenting term, we demonstrate the method on test problems. Our numerical experiments indicate that it is more favourable to use an augmenting term of an exponential form rather than the classical 1 or 2norm forms.
Introduction to Canonical Duality Theory
, 2009
"... Canonical Duality Theory is a versatile and potentially powerful methodology which is composed mainly of a canonical dual transformation, a complementarydual principle, and an associated triality theory. The canonical dual transformation can be used for modeling complex systems and to formulate ..."
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Canonical Duality Theory is a versatile and potentially powerful methodology which is composed mainly of a canonical dual transformation, a complementarydual principle, and an associated triality theory. The canonical dual transformation can be used for modeling complex systems and to formulate perfect dual problems without duality gap; the complementarydual principle presents a unified analytic solution form for general problems in continuous and discrete systems; the triality theory is comprised by a saddle minmax duality and two pairs of doublemin, doublemax dualities. This theory reveals an intrinsic duality pattern in complex phenomena and can be used to solve a very large class of challenging problems in complex systems. This lecture presents, within a unified framework, a selfcontained comprehensive introduction and some new developments on canonical duality theory for complex systems with emphasis on methods and applications in nonlinear analysis and optimization. Intrinsic relations among the popular semipositive programming, semiinfinite programming, complementarity theory, variational inequality, penalty methods, and the Lagrangian duality theory are revealed within the unified framework of the canonical duality theory. Applications are illustrated by a class of challenging (NPhard) problems in global optimization and nonconvex analysis. It is shown that by the use of the canonical dual transformation, nonconvex constrained primal problems can be converted into certain simple canonical dual problems, which can be solved to obtain all extremal points, and NPhard discrete optimization problems can be transformed to certain minimal stationary problems in continuous space. Optimality conditions (both local and global) for these extrema can be identified by the triality theory. This lecture brings some fundamentally new insights into nonconvex analysis, global optimization, and computational science.