Experience in global optimization shows that the computational time for solving a nonconvex problem usually grows exponentially with the number of variables. However not all the variables play an equal part in the “curse of dimensionality”. Variables which enter the problem in a convex way, i.e. such that the problem becomes convex when all the other variables are fixed, are often relatively “easy”. The main source of difficulty comes from the “nonconvex variables”, so that if these are few, then in many circumstances the problem is amenable to efficient solution methods, even though the overall dimension may be fairly large. Problems with few nonconvex variables or which become so after a linear transformation of variables, are called low rank nonconvex problems. In this Chapter we propose to study a wide class of low rank nonconvex problems characterized by a monotonicity property of functions with respect to a polyhedral cone having a nontrivial lineality.
KeywordsFeasible Point Affine Function Outer Approximation Good Feasible Solution Recession Cone
Unable to display preview. Download preview PDF.