The infinite dimensional Lagrange multiplier rule for convex optimization problems
The author provides optimality and duality assertions for the abstract convex optimization problem \[ f(x)\Rightarrow\min\text{ such that }x\in K=\{x\in S\mid g(x)\in-C,\, h(x)= 0_Z\}, \] where \(X\) is a linear topological space, \(Y\) and \(Z\) are real normed spaces with duals \(Y^*\) and \(Z^*\), \(Y\) is partially ordered by the convex cone \(C\subset Y\), \(f: X\to \mathbb R\) and \(g: X\to Y\) are convex functions, \(h: X\to Z\) is an affine-linear function and \(S\subset X\) is a convex set. The Lagrange dual problem is given by \[ \text{inf}\{L(x,u,v)\mid x\in S\}\Rightarrow\max\text{ such that } u\in C^*,\;v\in Z^*, \] where \(L(x,u,v)= f(x)+ \langle u,g(x)\rangle+\langle v,h(x)\rangle\) is the Lagrangian function and \(C^*\subset Y^*\) is the nonnegative dual cone of \(C\). Using the important ``Assumption S'' at the optimal solution \(x_0\in K\) according to \[ T_{\widetilde M}(0,0_Y,0_Z)\cap ((0,\infty)\times \{0_Y\}\times \{0_Z\})=\emptyset, \] where \(T_{\widetilde M}(0,0_Y,0_Z)\) is the classical contingent cone to the upper image set \[ \widetilde M= \{(f(x)- f(x_0)+ \alpha, g(x)+ y, h(x))\mid x\in S\setminus K, \alpha\geq 0, y\in C\} \] at the point \((0,0_Y,0_Z)\), it is pointed out that there exist \(u_0\in C^*\), \(v_0\in Z^*\) such that: (1) \((u_0,v_0)\) is a solution of the dual problem and strong duality holds, (2) \((x_0,u_0,v_0)\) is a saddle-point of the Lagrangean function, (3) if \(f\), \(g\) and \(h\) are sufficient smooth then \[ (f'(x_0)+\langle u,g'(x_0)\rangle+\langle v,h'(x_0)\rangle)(x- x_0)\geq 0,\quad\forall x\in S. \] Moreover, the three conditions are also sufficient for the ``Assumption S''. At the end of the paper, an application is presented, in particular the existence of Lagrange multipliers associated to the bi-obstacle problem is obtained.
scientific article; zbMATH DE number 5956033
scientific article