# Optimization with Multivalued Mappings: Theory, Applications by Stephan Dempe, Vyacheslav Kalashnikov By Stephan Dempe, Vyacheslav Kalashnikov

This e-book specializes in the large improvement that has taken position lately within the box of of nondifferentiable nonconvex optimization. assurance comprises the formula of optimality stipulations utilizing other forms of generalized derivatives for set-valued mappings (such as, for instance, the co-derivative of Mordukhovich), the outlet of latest purposes (the calibration of water offer systems), and the elaboration of recent resolution algorithms (e.g., smoothing methods).

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Xr = 1, (j) (j) (j) xq+1 = xq+2 = . . = x|A1 | = 0. Leaving the ﬁxed variables as they are, create r−p new shortest path problems that must satisfy the additional conditions xp+1 = 0, xp+1 = 1, xp+2 = 0, .. xp+1 = xp+2 = . . = xr−1 = 1, xr = 0. 4 A block sequential heuristic (BLOSH) While, as we shall see in the next section, the three MIP reformulations and the exact multipath algorithm allow to tackle medium size problems, the NPhard nature of TOP will ultimately limit the size of problems that can be solved to prove optimality.

22) 0 Hence, condition (21) implies that the vector F(x ) lies within the interior of the dual to the recession cone of the set X. 1 () The variational inequality problem: to ﬁnd a vector z ∈ X such that ∀x ∈ X, (23) (x − z)T F(z) ≥ 0 has a non-empty, compact, convex solution set. Proof. It is well-known  that the pseudo-monotonicity (20) and continuity of the mapping G imply convexity of the solution set Z = {z ∈ X : (x − z)T F(z) ≥ 0 ∀x ∈ X}, (24) of problem (23) provided that the latter is non-empty.

Since G is x) ≥ 0 for each z ∈ Z, continuous, the following limit relation holds: (z − x ¯)T G(¯ which means that x ¯ solves (29). Thus we have proved that every limit point of the generalized sequence Q solves BVI (23), (24), (29). Hence, Q can have at most one limit point. 2, it suﬃces to establish that the set Q is bounded, and consequently, the limit point exists. In order to do that, consider a norm-divergent sequence {xεk } of solutions to parametric problem (31) where εk → 0 as k → ∞. Without loss of generality, suppose (xεk − x0 ) → s ∈ Rn , s = 1; here x0 is the that xεk = x0 for each k, and xεk − x0 vector from condition (21).