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Convex Analysisの二,三の進展について

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Convex Analysis: (pms-28)

We introduce several solution concepts for multicriteria optimization problems, give a characterization of approximately efficient elements and discuss a general scalarization procedure. Furthermore, we derive necessary and sufficient optimality conditions, a minimal point theorem, a vector-valued variational principle of Ekeland’s type, Lagrangean multiplier rules and duality statements. An overview on vector variational inequalities and vector equilibria is given. Moreover, we discuss the results for special classes of vector optimization problems (vector-valued location and approximation problems, multicriteria fractional programming and optimal control problems).

Theory of Vector Optimization

Support Vector Machine is one of the classical machine learning techniques that can still help solve big data classification problems. Especially, it can help the multidomain applications in a big data environment. However, the support vector machine is mathematically complex and computationally expensive. The main objective of this chapter is to simplify this approach using process diagrams and data flow diagrams to help readers understand theory and implement it successfully. To achieve this objective, the chapter is divided into three parts: (1) modeling of a linear support vector machine; (2) modeling of a nonlinear support vector machine; and (3) Lagrangian support vector machine algorithm and its implementations. The Lagrangian support vector machine with simple examples is also implemented using the R programming platform on Hadoop and non-Hadoop systems.

Support Vector Machine

The paper contains a version of a minimax theorem with weakened convexity, extending a minimax theorem of Fan. The main result is obtained with the use of a generalized Gordan theorem, which is proved using a separation theorem. An example is also discussed.

A generalization of a minimax theorem of Fan via a theorem of the alternative

Dual characterizations of the containment of a closed convex set, defined by infinite convex constraints, in an arbitrary polyhedral set, in a reverse-convex set, defined by convex constraints, and in another convex set, defined by finite convex constraints, are given. A special case of these dual characterizations has played a key role in generating knowledge-based support vector machine classifiers which are powerful tools in data classification and mining. The conditions in these dual characterizations reduce to simple nonasymptotic conditions under Slater's constraint qualification.

Characterizing Set Containments Involving Infinite Convex Constraints and Reverse-Convex Constraints

In this paper some alternative theorems are presented for convexlike functions. These are then applied to obtain global and local optimality conditions for constrained minimization problems.

Convexlike alternative theorems and mathematical programming

In 1951, Fenchel discovered a special duality, which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions, using conjugates. Fenchel's duality is central to the study of constrained optimization. It requires an existence of an interior point of a convex set which often has empty interior in optimization applications. The well known relaxations of this requirement in the literature are again weaker forms of the interior point condition. Avoiding an interior point condition in duality has so far been a difficult problem. However, a non-interior point type condition is essential for the application of Fenchel's duality to optimization. In this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions. We also establish a necessary and sufficient condition for the e-subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions. Our results offer further insight into Fenchel's duality.

A new geometric condition for Fenchel's duality in infinite dimensional spaces

This paper examines nonsmooth constrained multi-objective optimization problems where the objective function and the constraints are compositions of convex functions, and locally Lipschitz and Gâteaux differentiable functions. Lagrangian necessary conditions, and new sufficient optimality conditions for efficient and properly efficient solutions are presented. Multi-objective duality results are given for convex composite problems which are not necessarily convex programming problems. Applications of the results to new and some special classes of nonlinear programming problems are discussed. A scalarization result and a characterization of the set of all properly efficient solutions for convex composite problems are also discussed under appropriate conditions.

Vaithilingam Jeyakumar

Papers

A generalization of a minimax theorem of Fan via a theorem of the alternative

Characterizing Set Containments Involving Infinite Convex Constraints and Reverse-Convex Constraints

Convexlike alternative theorems and mathematical programming

A new geometric condition for Fenchel's duality in infinite dimensional spaces

Convex composite multi-objective nonsmooth programming