Globalizing a nonsmooth Newton method via nonmonotone path searchBütikofer, S Bütikofer, S (2008). Globalizing a nonsmooth Newton method via nonmonotone path search. Globalizing a nonsmooth Newton method via nonmonotone path search AbstractWe give a framework for the globalization of a nonsmooth Newton method. In part one we start with recalling B. Kummer's approach to convergence analysis of a nonsmooth Newton method and state his results for local convergence. In part two we give a globalized version of this method. Our approach uses a path search idea to control the descent. After elaborating the single steps, we analyze and prove the global convergence resp. the local superlinear or quadratic convergence of the algorithm. In the third part we illustrate the method for nonlinear complementarity problems. Abstract We give a framework for the globalization of a nonsmooth Newton method. In part one we start with recalling B. Kummer's approach to convergence analysis of a nonsmooth Newton method and state his results for local convergence. In part two we give a globalized version of this method. Our approach uses a path search idea to control the descent. After elaborating the single steps, we analyze and prove the global convergence resp. the local superlinear or quadratic convergence of the algorithm. In the third part we illustrate the method for nonlinear complementarity problems.
In bilevel optimization there are two decision makers, commonly denoted as the leader and the follower, and decisions are made in a hierarchical manner: the leader makes the first move, and then the follower reacts optimally to the leader's action. It is assumed that the leader can anticipate the decisions of the follower, hence the leader optimization task is a nested optimization problem that takes into consideration the follower's response.In this talk we focus on new branch-and-cut (B&C) algorithms for dealing with mixed-integer bilevel linear programs (MIBLPs). We first address a general case in which intersection cuts are used to cut off infeasible solutions. We then focus on a subfamily of MIBLPs in which the leader and the follower share a set of items, and the leader can select some of the items to inhibit their usage by the follower. Interdiction Problems, Blocker Problems, Critical Node/Edge Detection Problems are some examples of optimization problems that satisfy the later condition. We show that, in case the follower subproblem satisfies monotonicity property, a family of "interdiction-cuts" can be derived resulting in a more efficient B&C scheme.These new B&C algorithms consistently outperform (often by a large margin) alternative state-of-the-art methods from the literature, including methods that exploit problem specific information for special instance classes.
In [S. Bütikofer, Math. Methods Oper. Res., 68 (2008), pp. 235-256] a nonsmooth Newton method globalized with the aid of a path search was developed in an abstract framework. We refine the convergence analysis given there and adapt this algorithm to certain finite dimensional optimization problems with C 1,1 data. Such problems arise, for example, in semi-infinite programming under a reduction approach without strict complementarity and in generalized Nash equilibrium models. Using results from parametric optimization and variational analysis, we work out in detail the concrete Newton schemes and the construction of a path for these applications and discuss a series of numerical results for semi-infinite and generalized semi-infinite optimization problems.then, under strong regularity at some solution for given x = x 0 , f i is locally (near x 0 ) a C 1,1 -function but does not belong to the class C 2 in general. A trivial but typical example showing this lack of smoothness is the parametric convex quadratic program min z {z 2 | z ≤ x} with optimal value f i (x) = (min{x, 0}) 2 . In our applications, we will especially consider lower level problems leading to optimal value functions with directionally differentiable gradients.Optimization problems of the type (1)-(2) appear in a quite natural way for bilevel models in optimization. In section 3 we study in detail C 1,1 optimization settings of (generalized) semi-infinite problems without strict complementarity in the lower level problem, and we sketch an optimization reformulation of normalized Nash equilibria. This suggests the use of nonsmooth Newton methods. In this paper we model the Karush-Kuhn-Tucker (KKT) conditions of the program (1) as a system of equations in Kojima's [27] (normal map) form. The simple structure of these functions allows A NONSMOOTH NEWTON METHOD WITH PATH SEARCH 2383 us to compute several standard generalized derivatives; we will mainly work with classical directional derivatives. This approach differs from other studies of Newtontype methods for solving C 1,1 programs, (generalized) semi-infinite programs, and Nash equilibrium problems in semismooth settings; see, e.g., [39,14,38,15,42,49,50,52,53,16].The implementation of the method NMPS was done with MATLAB and GAMS. Numerical experiments will be reported in section 4 for a series of test examples in semi-infinite and generalized semi-infinite programming, given in [2,48,49,50] and including certain robust optimization and design centering problems.We conclude this section by introducing some notation. By h ∈ C k (X, R d ) for k = 1, 2 (briefly h ∈ C k ) we indicate that h is a k-times continuously differentiable function from X ⊂ R n to R d . As introduced above, the symbol h ∈ C 1,1 is used analogously, while h ∈ C 0,1 means that h is locally Lipschitz. Dh(x) and D 2 h(x) denote the Jacobian matrix and Hessian matrix, respectively, if they exist. We say that an assertion holds for all x near x 0 if it holds for all x in a neighborhood U of x 0 . Givenwhere Θ h is the set of all p...
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