Introduction to Nonsmooth Optimization

Abstract

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Diagonal Bundle Method for Nonsmooth Sparse Optimization

“…The aggregation procedure gives us a possibility to retain the global convergence without solving the quite complicated quadratic direction finding problem (see, e.g., [38]) appearing in standard bundle methods. Note that the aggregate values are computed only if the last step was a null step.…”

“…However, to compute the search direction, the limited memory bundle method uses a dense approximation to the variable metric matrix and, thus, it fails to solve some sparse problems (see, e.g., [38,39]). …”

Proximal Bundle Method for Nonsmooth and Nonconvex Multiobjective Optimization

“…First we formulated several f • -pseudoconvex objective functions. Next we combined f • -pseudoconvex functions with classical convex test examples from [1]. Finally some nonconvex test examples [1] being not f • -pseudoconvex nor f • -quasiconvex were solved.…”

“…Next we combined f • -pseudoconvex functions with classical convex test examples from [1]. Finally some nonconvex test examples [1] being not f • -pseudoconvex nor f • -quasiconvex were solved. Furthermore, some f • -quasiconvex constraint functions were used in all the test examples.…”

“…In order to illustrate the functioning of MPB in more detail we consider the following problem: where f 1 is clearly f • -pseudoconvex (see [1]), f 2 is convex and g convex and thus f • -quasiconvex. We used first the starting point x 1 = (−0.5, −0.5) and the solution iteration by iteration is reported in Table 2.…”