Local and Global Optima © 2011 Daniel Kirschen and University of Washington

Which one is the real maximum? f(x) A D x © 2011 Daniel Kirschen and University of Washington

Which one is the real optimum? x2 A D x1 B C © 2011 Daniel Kirschen and University of Washington

Local and Global Optima The optimality conditions are local conditions They do not compare separate optima They do not tell us which one is the global optimum In general, to find the global optimum, we must find and compare all the optima In large problems, this can be require so much time that it is essentially an impossible task © 2011 Daniel Kirschen and University of Washington

Convexity If the feasible set is convex and the objective function is convex, there is only one minimum and it is thus the global minimum © 2011 Daniel Kirschen and University of Washington

Examples of Convex Feasible Sets x1min x1max © 2011 Daniel Kirschen and University of Washington

Example of Non-Convex Feasible Sets x1b x1c x1d © 2011 Daniel Kirschen and University of Washington

Example of Convex Feasible Sets A set is convex if, for any two points belonging to the set, all the points on the straight line joining these two points belong to the set x1 x2 x1 x2 x1 x2 x1 x1min x1max © 2011 Daniel Kirschen and University of Washington

Example of Non-Convex Feasible Sets x1b x1c x1d © 2011 Daniel Kirschen and University of Washington

Example of Convex Function f(x) x © 2011 Daniel Kirschen and University of Washington

Example of Convex Function © 2011 Daniel Kirschen and University of Washington

Example of Non-Convex Function f(x) x © 2011 Daniel Kirschen and University of Washington

Example of Non-Convex Function D x1 B C © 2011 Daniel Kirschen and University of Washington

Definition of a Convex Function f(x) z f(y) xa y xb x A convex function is a function such that, for any two points xa and xb belonging to the feasible set and any k such that 0 ≤ k ≤1, we have: © 2011 Daniel Kirschen and University of Washington

Example of Non-Convex Function f(x) x © 2011 Daniel Kirschen and University of Washington

Importance of Convexity If we can prove that a minimization problem is convex: Convex feasible set Convex objective function Then, the problem has one and only one solution Proving convexity is often difficult Power system problems are usually not convex There may be more than one solution to power system optimization problems © 2011 Daniel Kirschen and University of Washington