Duality for non-convex variational principles
The author discusses different ways of constructing a dual problem to the problem of minimizing a given function f(x) on a locally convex space X. Any function \(L(x,y^*)\) such that \(f(x)=\sup_{y^*\in Y^*}L(x,y^*)\) is called a Lagrangian of type I for the problem \(\min_{x\in X}f(x)\). Together with the classical dual problem in which the function \(g(y^*)=\inf_{x}L(x,y^*)\) is maximized on the conjugate space \(Y^*\), the author considers an ''anomalous'' dual problem in which one minimizes on \(Y^*\) the function \(h(y^*)=\sup_{x}L(x,y^*)\). This makes sense because, as shown by the author, sometimes the two numbers \(\inf_{x}\sup_{y^*} L(x,y^*)\) and \(\inf_{y^*}\sup_{x}L(x,y^*)\) are equal. The function \(L(x,y^*)\) is said to be a Lagrangian of type II for the problem \(\min_{x}f(x)\) if \(f(x)=\inf_{y^*} L(x,y^*)\). In case such a Lagrangian is given, a dual problem can be constructed which consists in minimizing the function \(g(y^*)\) on \(Y^*\). All these duality problems are illustrated in various situations including the case when \(f(x)=F(x,Ax),\) where \(F:X\times Y\to R\) is a given real-valued function and A:\(X\to Y\) is a given continuous linear operator. Much attention is given to the critical points of Lagrangians and to Hamilton's equations. The last five sections (our of 12) are devoted to particular problems (and their duals) connected with: Rayleigh principle, convex programming, semilinear operator equations, classical dynamical systems and nonlinear elliptic problems.
scientific article; zbMATH DE number 3846317