For solving nonsmooth systems of equations, the Levenberg-Marquardt method and its variants are of particular importance because of their locally fast convergent rates. Finitely manymaximum functions systems are very useful in the study of nonlinear complementarity problems, variational inequality problems, Karush-Kuhn-Tucker systems of nonlinear programming problems, and many problems in mecha...

This paper deals with the minimization of a class of nonsmooth pseudolinear functions over a closed and convex set subject to linear inequality constraints. We establish several Lagrange multiplier characterizations of the solution set of the minimization problem by using the properties of locally Lipschitz pseudolinear functions. We also consider a constrained nonsmooth vector pseudolinear opt...

It is a common practice to approximate “complicated” functions with more friendly ones. In large-scale machine learning applications, nonsmooth losses/regularizers that entail great computational challenges are usually approximated by smooth functions. We re-examine this powerful methodology and point out a nonsmooth approximation which simply pretends the linearity of the proximal map. The new...

Practical optimization problems often involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such large problems are restricted to certain meaningful intervals. In the paper [Karmitsa, Mäkelä, 2009] we described an efficient limited memory bundle method for large-scale nonsmooth, possibly nonconvex, bound constrained optimization. Although this method wor...

We propose a new first-order splitting algorithm for solving jointly the primal and dual formulations of large-scale convex minimization problems involving the sum of a smooth function with Lipschitzian gradient, a nonsmooth proximable function, and linear composite functions. This is a full splitting approach, in the sense that the gradient and the linear operators involved are applied explici...

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M . We give several applications of this theory, concerning: (1) differentiability and geometrical properties of the distance function to a closed subset C of M ; (2) solvability and implicit function theorems for nonsmooth functions on M ; (3) conditions on the existence of...

This paper develops a Takagi-Sugeno fuzzy approach for modeling a DC-DC voltage-mode controlled buck converter as a nonlinear, nonsmooth system to capture all the essential fast-scale nonlinearities that occur at controller clock frequency. A tractable mathematical stability analysis, employing nonsmooth Lyapunov functions, is proposed for identifying converter fast-scale instabilities and the ...

A few Karush-Kuhn-Tucker type of sufficient optimality conditions are given in this paper for nonsmooth continuous-time nonlinear multi-objective optimization problems in the Banach space L∞ [0, T ] of all n-dimensional vector-valued Lebesgue measurable functions which are essentially bounded, using Clarke regularity and generalized convexity. Further, we establish duality theorems for Wolfe an...

According to the Morse-Sard theorem, any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse-Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization, while we restrict the functions under consideration to be semi-algebraic...

In this work we continue the nonsmooth analysis of absolutely symmetric functions of the singular values of a real rectangular matrix. Absolutely symmetric functions are invariant under permutations and sign changes of its arguments. We extend previous work on subgradients to analogous formulae for the proximal subdifferential and Clarke subdifferential when the function is either locally Lipsc...