We present for the reader's attention a special issue of Applicable Analysis devoted to selected topics in modern variational analysis and its applications to various problems of optimization, equilibria, control of systems governed by ordinary differential and partial differential equations, stochastic processes, etc.
Modern variational analysis has been well recognized as an active area of mathematics with the emphasis on applications. On one hand, it particularly concerns the study of optimization-related and equilibrium problems while, on the other hand, it applies variational/optimization principles as well as perturbation and approximation techniques to the analysis of a broad spectrum of problems that may not be of a variational nature.
This area of applied mathematics can be treated as an outgrowth of the classical calculus of variations, optimal control theory, and constrained optimization with a special attention to sensitivity/stability analysis with respect to perturbations.
Among characteristic features of modern variational analysis is a strong involvement of mathematical objects with nonstandard, nonsmooth structures (e.g., nondifferentiable functions, set with nonsmooth boundaries, and set-valued mappings), which frequently arise in the frameworks of optimization, equilibria, and systems control being in fact naturally generated by the usage of advanced variational principles and perturbation techniques. Furthermore, powerful variational principles in applied sciences (particularly in physics, mechanics, biology, and economics) also give rise to nonsmooth structures and often motivate the growth of new form of analysis. All these phenomena require the development and applications of appropriate tools of generalized differentiation.
This special issue consists of 12 papers dealing with both qualitative and numerical aspects of variational analysis and its applications to optimization and control. Let us briefly describe the main attitudes and achievements of each paper included in this issue at the alphabetical order.
The paper by Al-Homidan, Ansari and Yao is devoted to the investigation of collectively fixed point theorem and maximal element theorem for a family of multivalued maps in the setting of topological semilattice spaces. As an application of their maximal element theorem, the authors prove the existence of solutions of generalized abstract economies with two constraint correspondences. They also consider the system of vector quasi-equilibrium problems and system of generalized vector quasi-equilibrium problems and present some applications of their results to constrained Nash equilibrium problem for vector-valued functions with infinite number of players as well as to semi-infinite programs.
The paper by Arutyunov and Zhukovskiy concerns control systems governed by ordinary differential equations with mixed constraints on control and state variables. Problems with constraints of this type are among the most difficult in control theory. The authors mainly concentrate on solvability of the control problem under consideration with respect to mixed constraints. They consider separately the smooth case with the usage of the so-called 2-regularity constraint qualification and then the general nonsmooth case in which they apply a new version of the covering/metric regularity property (fundamental in variational analysis and its applications) and an appropriate variant of the implicit function theorem.
The paper by Calin, Chang and Hu addresses the study of broad classes of continuous stochastic processes described via either Brownian motion or via martingales of special types. The authors mainly consider optimization problems related to best mean square approximations and employ modern variational methods for their solutions. They implement the results obtained to various examples, including the classical and new ones, and present simulations for each example under consideration.
The main attention of the paper by Caraballo is paid to developing local linear approximations to the anisotropic surface energy generated by finite perimeter partitions of finite-dimensional spaces and convex surface energy density functions associated with such partitions. The author proposes a new notion of locally simple partitions, which is useful for the study of many variational problems arising in materials science, image processing, and other fields. Employing advanced variational techniques, he establishes several key partial regularity theorems and a covering theorem for locally simple partitions in a rather general setting. Refined error bounds are also derived in this way.
The paper by Chen, Huang and Yang is devoted to problems of multiobjective/vector optimization and mainly focuses on numerical aspects of their variational analysis. The authors develop new algorithms of the proximal point type to find weak Pareto optimal solutions in finite-dimensional case. They algorithms and convergence analysis are based on extended versions of vector Bregman functions, which have been proved to provide efficient tools of numerical analysis in scalar problems of convex constrained optimization.
The major attention of the paper by Henrion and Surowiec is paid to remarkable classes of hierarchical optimization problems related to bilevel programming and mathematical programs with equilibrium constraints. These classes of optimization problems are intrinsically nonsmooth regardless of the differentiability of their initial data. Using advanced tools of variational analysis and generalized differentiation, the authors derive new optimality conditions under mild constraint qualifications. The results obtained are applied to a bilevel program arising from a discretize obstacle control problem governed by partial differential equations of the elliptic type, which is of high importance for its own sake.
The paper by Lasiecka, Triggiani and Zhang concerns a practically motivated mini–max model describing an interaction between control and disturbance in fluid mechanics. The dynamics in this fluid-structure interaction model is governed by hyperbolic-parabolic equations, where the structure is described by the hyperbolic system of dynamic elasticity and the fluid is modeled by the linearized Navier-Stockes equations of the parabolic type. Developing a delicate variational analysis, the authors overcome the main mathematical difficulties arising from the mismatch between the hyperbolic and parabolic dynamics in the coupled system and establish in this way verifiable conditions ensuring the solvability of the game-theoretical problem under consideration.
The paper by Lee and Yen is devoted to the core theoretical issues of variational analysis related to the study of implicit multifunctions via generalized differentiation. In these lines of development, the authors derive new upper and lower estimates for coderivatives of set-valued mapping and derive verifiable necessary as well as sufficient conditions for good (Lipschitz-like, metric regularity) properties of implicit multifunctions defined by some generalized equations/variational conditions. They also present a number of examples and counterexamples illustrating the most important features of the results obtained.
Another paper on multiobjective optimization and equilibria are presented by Luc, Rocca and Papalia. The authors mainly concentrate on multicriteria supply-demand network problems and study equilibrium notions of the Nash-Wardrop type. They reveal striking differences between usual and weak notions of the vector equilibrium under consideration and reduce both of them to the corresponding vector variational inequalities. To study the latter class of optimization-related problems, the authors develop a nice scalarization technique, which allows them to characterize vector equilibrium via Pareto efficiency.
The paper by Merino, Neitzel and Trölzsch provides qualitative and numerical analysis of an interesting and largely underinvestigated class of optimal control problems governed by elliptic partial differential equations with pointwise state constraints and finitely many control parameters. The authors show that this problem, under an appropriate discretization, can be reduced to a family of semi-infinite programs. The latter class of optimization problems is among the most attractive and challenging from the viewpoint of modern variational analysis. In this way, the authors derive necessary and sufficient conditions for the original PDE control problem and employ them to conduct an elegant and efficient numerical analysis with establishing convergence and precise error bounds.
The paper by Mordukhovich is devoted to optimal control and feedback design of constrained parabolic systems in uncertainty conditions. The problems under consideration are among the most challenging in control theory being largely motivated by applications. Using advanced techniques of variational analysis, which involve a number of approximation procedures and exploit asymptotic properties of parabolic systems on the infinite horizon, the author obtains verifiable characteristics of optimal controls under the worst perturbations and employs them to compute optimal parameters of discontinuous feedback regulators ensuring robust stability of the closed-loop system under any admissible perturbations.
The final paper of this special issue written by Voisei and Zălinescu addresses the so-called complementarity gap functions and their applications to optimization. The paper analyzes some results of Gao and Strang in this direction and reveal several errors and inconsistencies in their formulations and proofs. Furthermore, the authors present corrections of the aforementioned results under additional assumptions and provide new proofs based on the techniques of convex analysis in infinite-dimensional spaces.
Acknowledgements
Research for Der-Chen Chang was partially supported by the US National Science foundation grant DMS-0631541 and by the Hong Kong grant RGC-600607. Research for Boris S. Mordukhovich was partially supported by the US National Science Foundation under grant DMS-0603846 and by the Australian Research Council under grant DP-12092508. Research for Jen-Chih Yao was partially supported by the NSC grant 98-2923-E-110-003-MY3.