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The least squares support vector machine (LS-SVM) is an effective method to deal with classification and regression problems and has been widely studied and applied in the fields of machine learning and pattern recognition. The learning algorithms of the LS-SVM are usually conjugate gradient (CG) and sequential minimal optimization (SMO) algorithms. Based on this, we propose a conjugate functional gain SMO algorithm and theoretically prove its asymptotic convergence. This algorithm combines the conjugate direction method and the functional gain SMO algorithm with second-order information, which increases the functional gain of the plain SMO algorithm. In addition, we also provide a generalized SMO-type algorithm framework with a simple iterative format and easy implementation for other LS-SVM training algorithms. The numerical results show that the execution time of this algorithm is significantly shorter than that of the other plain SMO-type algorithms and CG-type algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush-Kuhn-Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented. [ABSTRACT FROM AUTHOR]

In Part I, sufficient and necessary optimality conditions and the image regularity conditions of constrained scalar and vector extremum problems are reviewed for Image Space Analysis. Part II presents the main feature of the duality and penalization of constrained scalar and vector extremum problems by virtue of Image Space Analysis. In the light, as said in Part I and Part II, to describe the state of Image Space Analysis for constrained optimization, and to stress that it allows us to unify and generalize the several topics of Optimization, in this Part III, we continue to give an exhaustive literature review on separation functions, gap functions and error bounds for generalized systems. Part III also throws light on some research gaps and concludes with the scope of future research in this area. [ABSTRACT FROM AUTHOR]

In the light, as said in Part I, of showing the main feature of image space analysis—to unify and generalize the several topics of optimization—we continue, in Part II, considering duality and penalization. In the literature, they are distinct sectors of optimization and, as far as we know, they have nothing in common. Here it is shown that they can be derived by the same “root.” [ABSTRACT FROM AUTHOR]

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of vector topical function. It contains some other categories of topical functions as special cases and can be interpreted as weak separation functions in image space analysis. We establish its envelope result and investigate its properties in the frame of abstract convexity. Then, we present the corresponding conjugation and subdifferential, and observe the relationships among these concepts. Finally, as applications, we obtain some dual results for some vector optimization, where the object is expressed as the difference of vector topical functions. [ABSTRACT FROM AUTHOR]

In this paper, we propose weak separation functions in the image space for general constrained vector optimization problems on strong and weak vector minimum points. Gerstewitz function is applied to construct a special class of nonlinear separation functions as well as the corresponding generalized Lagrangian functions. By virtue of such nonlinear separation functions, we derive Lagrangian-type sufficient optimality conditions in a general context. Especially for nonconvex problems, we establish Lagrangian-type necessary optimality conditions under suitable restriction conditions, and we further deduce Karush-Kuhn-Tucker necessary conditions in terms of Clarke subdifferentials. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend the concepts of linearizing cone, regularity assumption and Lagrange multiplier rule due to Maurer and Zowe [H. Maurer, J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Program. 16 (1979) 98–110] to an optimization problem under inclusion constraints. By virtue of the Robinson–Ursescu open mapping theorem, we obtain a Kuhn–Tucker necessary optimality condition. Moreover, we propose a Lagrangian by using the support function for set-valued maps, and establish some second-order sufficient and necessary optimality conditions for a strict local minimizer of order 2 based on the second-order derivative of the Lagrangian. As applications, we also investigate some second-order optimality conditions for the strict local minimizer of order 2 of a smooth scalar optimization problem with equality and inequality constraints. [Copyright &y& Elsevier]