The goal of this article is to provide first-order necessary conditions of optimality for every local optimal pair (x~*, u~*) of the problem Minimize L(x, u), on all (x, u) ∈X*U subject to Ax-Bu f, where X, U, Y are Banach spaces, f ∈ Y, A: D(A) is contained in X → 2~Y, B:D(B) is contained → 2~Y are linear closed operators, L : X * U → IR is locally Lipschitz continuous or of the form L(x,u) = g(x) + h(u) where g:X → IR is locally Lipschitz continuous and h : U → IR∪{∞} is proper and convex. More precisely, the optimality conditions for (x~*, u~*) are partial deriv_xL(x~*, u~*) + A~* p 0, partial deriv_yL(x~*,u~*) -B~*q 0, p - q ∈(R(A) - R(B))~⊥, for some p,q ∈Y~*, Here Partial derivL(x~*, u~*) = (partial deriv_xL(x~*, u~*), partial deriv_yL(x~*,u~*)). Under additional hypothese these necessary conditions become: partial deriv_xL(x~*,u~*) + A~* p 0, partial deriv_yL(x~*,u~*) - B~*p 0.
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机译：本文的目的是为问题的每个局部最优对（x〜*，u〜*）提供一阶必要的最优性条件在所有（x，u）∈X*上使L（x，u）最小化U受Ax-Bu f约束，其中X，U，Y是Banach空间，f∈Y，A：X中包含D（A）→2〜Y，B：D（B）中包含→2〜Y是线性封闭算子，L：X * U→IR局部为Lipschitz连续或形式为L（x，u）= g（x）+ h（u）其中g：X→IR局部为Lipschitz连续，h：U→ IR∪{∞}是适当的且凸的。更准确地说，（x〜*，u〜*）的最优条件是偏导数xL（x〜*，u〜*）+ A〜* p 0，偏导数yL（x〜*，u〜*）-B〜* q 0，p-q∈（R（A）-R（B））〜⊥，对于某些p，q∈Y〜*，这里Partial derivL（x〜*，u〜*）=（partial deriv_xL（x〜 *，u〜*），部分deriv_yL（x〜*，u〜*））。在另外的假设下，这些必要条件变为：部分deriv_xL（x〜*，u〜*）+ A〜* p 0，部分deriv_yL（x〜*，u〜*）-B〜* p 0。
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