By Boris S. Mordukhovich
Entire and state-of-the artwork learn of the fundamental thoughts and rules of variational research and generalized differentiation in either finite-dimensional and infinite-dimensional areas provides quite a few purposes to difficulties within the optimization, equilibria, balance and sensitivity, keep an eye on concept, economics, mechanics, and so forth.
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Additional info for Variational Analysis and Generalized Differentiation II: Applications (Grundlehren der mathematischen Wissenschaften)
Example text
26) ∇si (¯ when ϕi are locally Lipschitzian for i = m + 1, . . , m + r , it is sufficient to observe that f is automatically SNC at x¯ in this case and then to apply the x ), which gives scalarization formula to the coderivative D ∗ f (¯ m+r D ∗ f (¯ x )(λm+1 , . . , λm+r ) = ∂ x) . 1 Necessary Conditions in Mathematical Programming 25 It remains to prove (ii). 11(iii) with F = f := (ϕ1 , . . 24). 28) under the general assumptions made in (ii) with some x )(λ1 , . . , λm+r ). When all ϕ1 , . .
M+r ) ∈ IR m+r αi ≤ 0 for i = 1, . . 24) αi = 0 for i = m + 1, . . , m + r . 24). 12). 23). 23) that follow from the results of Subsect. 2. As always, we use the common coderivative symbol D ∗ for the basic coderivatives of mappings with values in finite-dimensional spaces. For brevity we present only necessary optimality conditions without constraint qualifications; the normal counterparts of these conditions either follow from the corresponding results of the preceding subsection or can be derived in a similar way.
Proof. To prove (i) under the general assumptions made, we take arbitrary x ) for i = 0, . . , m and apply the variational description elements xi∗ ∈ ∂ + ϕi (¯ x ). 88(ii) with S = LC 1 to the subgradients −xi∗ ∈ ∂(−ϕi )(¯ In this way we find functions si : X → IR for i = 0, . . , m satisfying x ) = ϕi (¯ x ) and si (x) ≥ ϕi (x) around x¯ si (¯ x ) = xi∗ . 23) but with the cost and inequality constraint functions continuously differentiable around x¯: 24 5 Constrained Optimization and Equilibria ⎧ minimize s0 (x) subject to ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ si (x) ≤ 0, i = 1, .