LECTURE 16. Course: “Design of Systems: Structural Approach” Dept. “Communication Networks &Systems”, Faculty of Radioengineering & Cybernetics Moscow Institute of Physics and Technology (University) / Mark Sh. Levin Inst. for Information Transmission Problems, RAS Oct. 8, 2004 PLAN: 1.Technical documentation (Russian experience as a set of basic documents) 2.Types of interchange techniques 3.Genetic algorithms 4.Multi-objective evolutionary optimization 5.Multidisciplinary optimization 6.Mixed Integer Nonlinear Programming

Technical documentation (Russian experience) 1.Preliminary “avan”-project 2.”Avan”-project 3.Technical suggestion (proposal) 4.Technical project 5.Work-project 6.Report on the 1 st -stage of utilization (including suggestion on system improvement) 7.Resultant report on utilization (including suggestion on system improvement) BASIC VERSION 1.Technical suggestion (proposal) 2.Technical-work project 3.Report on utilization COMPESSED VERSION

Types of 2-Exchange Technique (Illustration) jk Sequence... jj+1 Sequence... j n Sequence... BASIC VERSION VERSION FOR NEIGHBOR ELEMENTS VERSION WITH LAST (1-ST) ELEMENT

3-Exchange Technique & 4-Exchange Technique (Illustration) j-2 j-1j Sequence... j+1 VERSION FOR 3-EXCHANGE CASE j-1j Sequence... j+1 VERSION FOR 4-EXCHANGE CASE

Exchange Technique: Two-dimensional Case (Illustration) Array... 2-EXCHANGE CASE... Array... 4-EXCHANGE CASE...

Genetic algorithms (illustration for knapsack problem) Solution x 0 = ( x 1, …, x i, …, x m ) Basic knapsack problem is: max  m i=1 c i x i s.t.  m i=1 a i x i  b x i  {0, 1}, i = 1, …, m

Genetic algorithms (illustration for knapsack problem) STEP 1. AN INITIAL SOLUTION x 0 STEP 2. DIVIDING OF x 0 INTO 2 PARTS STEP 3. MUTATION (GENERATION OF VERSIONS ): *exchange of elements, *change of elements, etc.... a 1 a 2 a 3 a 4... b 1 b 2 b 3 b 4... STEP 4. GENERATION OF NEW SOLUTIONS BY PAIRS ( a i, b j )... c 1 c 2 c 3 c 4... a b

Genetic algorithms (illustration for knapsack problem) STEP 5. Deletion of some solutions by resource constraint (  b )... c 1 c 2 c 3 c 4... STEP 6. Selection of the best solution(s)... c 1 c 2 c 3... Selection by two ways: 1.Selection by utility function 2.Selection of Pareto-effective solutions. This is Multi-Objective Evolutionary Optimization STEP 7. Repetion of steps 2, 3, 4, 5, and 6 for selected solutions

Multidisciplinary optimization (structural & aerospace engineering) max f (x) ( or extr f(x) ) subject to  1 (x)  W weight B 1   2 (x)  B 2 height C 1   3 (x)  C 2 temperature D 1   4 (x)  D 2 reliability...  k (x)  0 General optimization model with constraints corresponding to certain disciplines (e.g., weight, reliability, etc.):  j (x) is a constraint function (1  j  k) Int. Society for Structural and Multidisciplinary Optimization (civil engineering, ship engineering, marine engineering, aerospace engineering) // Optimal design of structures (including issues of fluids)

Mixed Integer Nonlinear Programming (process systems engineering & chemical engineering) min F ( x, y ) subject to h (x, y) = 0 g (x, y)  0 where x is vector of binary variables (selection of subsystems) y is vector of continuous variables / parameters (e.g., size) Generalized optimization model that involves integer & continuous variables: *Global optimization *Process Systems Engineering (chemical engineering, etc.) *Prof. C.A. Floudas (Princeton Univ, Chemical Engineering) *Prof. I.E. Grossmann (Carnegie Mellon Univ., Chemical Engineering)

Basic Methods for Mixed Integer Nonlinear Programming 1.Branch-And-Bound method 2.Combinatorial hybrid techniques 3.Gradient method 4.Interior point method