The Lagrange approach to infinite linear programs
Let \((X, Y)\) and \((Z, W)\) be two dual pairs of vector spaces and let \(X_+\) and \(Z_+\) be positive cones in \(X\) and \(Z\), respectively. Also, let \(Y_+:= X^*_+\subset Y\) and \(W_+:= Z^*_+\subset W\) the dual cones of \(X_+\) and \(Z_+\). The authors study a general, infinite-dimensional, inequality-constrained linear program \[ \text{IP:}\qquad \text{minimize }\{\langle x,c\rangle\mid Ax- b\in Z_+, x\in X_+\}, \] and the corresponding dual program \[ \text{IP}^*:\quad \text{minimize }\langle b,w\rangle\qquad\text{subject to:} -A^*w+ c\in Y_+,\;w\in W_+. \] They use the method of Lagrange multipliers to determine necessary and sufficient conditions for the strong duality of IP, which means that IP and \(\text{IP}^*\) are both solvable and that their optimal values coincide. The authors also show how the results for IP can be applied to programs with equality constraints and illustrate the results with applications to the general capacity and the mass transfer problems.
infinite-dimensional, inequality-constrained linear program
scientific article; zbMATH DE number 1740557
scientific article