David Yang Gao

Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization

This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in ℝn can be reformulated into certain smooth/convex unconstrained dual problems in ℝm with m ⩽ n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in \realn can be reformulated into certain smooth/convex unconstrained dual problems in \realm with m≤slant n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.

Perfect duality theory and complete solutions to a class of global optimization problems

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 so-called canonical dual transformation developed recently. It is shown that, by this method, these difficult nonconvex constrained primal problems in can be converted into a one-dimensional 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.

Canonical Duality Theory and Solutions to Constrained Nonconvex Quadratic Programming

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 {\open {R}}^{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 by-product. Several examples are illustrated.

Advances in mechanics and mathematics

List of Figures. Preface. Part I: Nonsmooth Mechanics of Solids. 1. Dynamics of Rigid Bodies Systems with Unilateral or Frictional Constraints P. Ballard. 2. Semilinear Hemivariational Inequalities D. Motreanu, Z. Naniewicz. Part II: Dendritic Growth in Fluids. 3. Dendritic Growth With Convection Jian-Jun Xu.

Solutions to quadratic minimization problems with box and integer constraints

This paper presents a canonical duality theory for solving quadratic minimization problems subjected to either box or integer constraints. Results show that under Gao and Strang’s general global optimality condition, these well-known nonconvex and discrete problems can be converted into smooth concave maximization dual problems over closed convex feasible spaces without duality gap, and can be solved by well-developed optimization methods. Both existence and uniqueness of these canonical dual solutions are presented. Based on a second-order canonical dual perturbation, the discrete integer programming problem is equivalent to a continuous unconstrained Lipschitzian optimization problem, which can be solved by certain deterministic technique. Particularly, an analytical solution is obtained under certain condition. A fourth-order canonical dual perturbation algorithm is presented and applications are illustrated. Finally, implication of the canonical duality theory for the popular semi-definite programming method is revealed.

Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications

where I ⊂R is an open interval, f(x) is a given function, is a nonlinear di erential operator, and W ( ) ∈ L(I) is a piecewise Gâteaux di erentiable function of = (u); Ua is a closed convex subspace of a re exive Banach space U. This general nonconvex, nonsmooth variational problem appears in many nonlinear systems. For example, in the nonlinear equilibrium problem of Ericksen’s bar subjected to axial extension [17], or the post-buckling analysis of extended nonlinear beam subjected to a compressed load [26], the nite strain = (u)= 2u 2 ; x − is a quadratic operator,

General Analytic Solutions and Complementary Variational Principles for Large Deformation Nonsmooth Mechanics

This paper presents a nonlinear dual transformation method and general complementary energy principle for solving large deformation theory of elastoplasticity governed by nonsmooth constitutive laws. It is shown that by using this method and principle, the nonconvex and nonsmooth total potential energy is dual to a smooth complementary energy functional, and fully nonlinear equilibrium equations in finite deformation problems can be converted into certain tensor equations. The algebraic relation between the first and the second Piola–Kirchhoff stresses are revealed. A closed form solution for general three-dimensional large deformation boundary value problems is obtained. The properties of this general solution are clarified by a triality extremum principle. This triality theory reveals an important phenomenon in nonconvex variational problems. Applications are illustrated by nonlinear, nonsmooth equilibrium problems in Henckys plasticity, 3D cylindrical structures and post buckling analysis of elastoplastic bar with jumping and hardening effects. The idea and methods presented in this paper can be used and generalized to solve many nonlinear boundary value problems in finite deformation theory.

Canonical duality theory: Unified understanding and generalized solution for global optimization problems

Canonical duality theory is a potentially powerful methodology, which can be used to model complex systems with a unified solution to a wide class of discrete and continuous problems in global optimization and nonconvex analysis. This paper presents a brief review and recent developments of this theory with applications to some well-know problems, including polynomial minimization, mixed integer and fractional programming, nonconvex minimization with nonconvex quadratic constraints, etc. Results shown that under certain conditions, these difficult problems can be solved by deterministic methods within polynomial times, and NP-hard discrete optimization problems can be transformed to certain minimal stationary problems in continuous space. Concluding remarks and open problems are presented in the end.

Finite deformation beam models and triality theory in dynamical post-buckling analysis☆

Abstract Two new finitely deformed dynamical beam models are established for serious study on non-linear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola–Kirchhoff’s type stress only) in finite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that if the gap function introduced by Gao and Strang in 1989 in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Duffing equations are obtained.

Nonlinear elastic beam theory with application in contact problems and variational approaches

From the first equation we know that a: is a constant, so this beam model is actually a linear ordinary differential equation. If the beam is quite thick, the deformation in lateral direction can not be ignored. By considering the stress in lateral direction, a nonlinear beam theory is developed in this paper for large displacement and small strain elastic beam theory. Application to the unilateral problem with obstacle is illustrated and a nonlinear complementarity problem is proposed. We proved that this nonlinear complementarity problem is equivalent to a variational inequality and a primal variational approach.