Research Update (10/08-10/22)

1. Research Update (10/08-10/22) • Research Done • Writing Thesis: 3 papers + 2 appendices • IDC paper • Overview of deterministic global optimization algorithm • Research to be done • completion of thesis (by oct 26th) • IDC paper review • AICHE poster preparation

2. Overview of deterministic global optimization 10/21/09 Jeonghwa Moon

3. Introduction • Global vs Local optimization • nonconvex vs convex • Deterministic vs Stochastic Method • Deterministic • mathematical techniques based on the concept that next step can be determined precisely from the past behavior of a set of data • guarantee e-global minimum in mathematical sense • Lipschitzian methods; BB procedures ; cutting plane methods; difference of convex functions and reverse convex methods; outer approximation approaches; primal-dual methods; reformulation-linearization, interval methods [Hansen, 1992]. • Stochastic : • incorporate with random numbers to predict next step • offer asymptotic convergence guarantees only at infinity for a very wide class of optimization problems • Simulated annealing, genetic algorithms, clustering methods convex nonconvex

4. aBB Frameworks • Proposed for locating the global minimum solution of constrained nonconvex problems by Floudas group [Androulakis et al., 1995; Adjiman et al, 1996]. • Based on a branch and bound framework and the modification of converging lower and upper bounds. • addresses nonconvex minimization problem and guarantees global optimality. • Lower bounds are obtained through the solution of convex programming problems and upper bounds based on the solution of nonconvex programming with local methods

5. Contents • Branch and bound framework • Problem definition • Convex relaxation • Whole procedure

6. Branch and Bound Framework

7. problem formulation

8. problem formulation • CO(x), Cj(x) : convex part • NCOk(x) : noncovex terms • and NCj(x) represent convex part and nonconvex part in constraints. • The last terms: bilinear terms i • All nonlinear equality constraints hj(x) = 0 in (1) have been replaced by two inequalities in (2).

9. f(x) upper bound f(x) L(x) lower bound xL xU L(x) xL xU * Convex relaxation • To convert nonconvex function to convex function convex term a :positive constant L(x) convex function f(x) original nonconvex function f(x) L(x) upper bound: current minimum point of f(x) lower bound : minimum point of L(x) xL xU a : too small, L(x) is nonconvex a : too big, UBD-LBD is too big

10. how to set a H(x) : Hessian matrix

11. Convex relaxation-continued (e)The underestimators constructed over supersets of the current set are always less tight than the underestimator constructed over the current for every point within the current box constraints. |U1-L1| < |U1-L0| |U1-L2| < |U1-L0| U1 L0

12. L2 L1 0 1 2 UB LB

13. L3 Fathom L4 L2 0 1 2 4 3 >UB

14. L4 L6 L5 Fathom 0 1 2 4 3 5 6 UB LB

15. L7 L8 Fathom Convergence 0 1 2 6 3 5 4 7 8

16. Branch and Bound

17. Next week • Proof of definition of a parameter • Special cases for convex relaxation • BARON vs aBB