Ghost Penalties in Nonconvex Constrained Optimization: Diminishing Stepsizes and Iteration Complexity

We consider nonconvex constrained optimization problems and propose a new approach to the convergence analysis based on penalty functions. We make use of classical penalty functions in an unconventional way, in that penalty functions only enter in the theoretical analysis of convergence while the algorithm itself is penalty free. Based on this idea, we are able to establish several new results, including the first general analysis for diminishing stepsize methods in nonconvex, constrained optimization, showing convergence to generalized stationary points, and a complexity study for sequential quadratic programming–type algorithms.

Nonconvex constrained optimization by a filtering branch and bound

A major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the $$\alpha $$ α BB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained optimization problem. Moreover, the multiobjective reformulation enables to identify the trade-off between constraint satisfaction and objective value which is also reflected in the quality guarantee. Numerical tests validate that we indeed can find feasible and often optimal solutions where the classical single-objective $$\alpha $$ α BB method fails, i.e., it terminates without ever finding a feasible solution.

Combined Emission Economic Dispatch of Power System in Presence of Solar and Wind Using Flower Pollination Algorithm

ED (Economic Dispatch) problem is one of the vital step in operational planning. It is a nonconvex constrained optimization problem. However, it is solved as convex problem by approximation of machine input/output characteristics, thus resulting in an inaccurate result. Reliable, secure and cheapest supply of electrical energy to the consumers is the prime objective in power system operational planning. Increase in fuel cost, reduction in fossil-fuel assets and ecological concerns have forced to integrate renewable energy resources in the generation mix. However, the instability of wind and solar power output affects the power network. For solution of such solar and wind integrated economic dispatch problems, evolutionary approaches are considered potential solution methodologies. These approaches are considered as potential solution methodologies for nonconvex ED problem. This paper presents CEED (Combined Emission Economic Dispatch) of a power system comprising of multiple solar, wind and thermal units using continuous and binary FPA (Flower Pollination Algorithm). Proposed algorithm is applied on 5, 6, 15, 26 and 40 thermal generators by integrating several solar and wind plants, for both convex and non-convex ED problems. Proposed algorithm is simulated in MATLAB 2014b. Results of simulations, when compared with other approaches, show promise of the approach.