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- hpa_score is calculated for each vnfc by adding the score parameter from hpa policy.
- If there are multiple VNFC's per VNF then the score of the best flavor selected is cumulatively added and stored as candidate[hpa_score].
- Translator should be able to identify hpa_score objective function from homing template.
- A new objective function needs to created in solver module. Objective function is computed using the candidate[hpa_score] parameter.

Lecture 9: Multi-Objective Optimization Suggested reading: K. Deb, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons, Inc., 2001 2 ? Involve more than one objective function that are to be minimized or maximized ? Answer is set of solutions that define the best tradeoff between competing objectives Multi-Objective Optimization Problems (MOOP) 3 General Form of MOOP ? Mathematically x x x i n h k K g j J f m M U i i L i k j m , 1 , 2 , , ( ) 0 , 1 , 2 , , subject to ( ) 0 , 1 , 2 , , min/max ( ), 1 , 2 , , ( ) ( ) L L L L ≤ ≤ = = = ≥ = = x x x lower bound upper bound 4 ? In the single-objective optimization problem, the superiority of a solution over other solutions is easily determined by comparing their objective function values ? In multi-objective optimization problem, the goodness of a solution is determined by the dominance Dominance 5 Definition of Dominance ? Dominance Test ? x 1 dominates x 2, if ? Solution x 1 is no worse than x 2 in all objectives ? Solution x 1 is strictly better than x 2 in at least one objective ? x 1 dominates x 2 x 2 is dominated by x 1 6 ? 1 Vs 2: 1 dominates 2 ? 1 Vs 5: 5 dominates 1 ? 1 Vs 4: Neither solution dominates Example Dominance Test f2 (minimize) f1 (maximize) 1 2 3 4 5 7 ? Non-dominated solution set ?Given a set of solutions, the non-dominated solution set is a set of all the solutions that are not dominated by any member of the solution set ? The non-dominated set of the entire feasible decision space is called the Pareto-optimal set ? The boundary defined by the set of all point mapped from the Pareto optimal set is called the Paretooptimal front Pareto Optimal Solution 8 Graphical Depiction of Pareto Optimal Solution feasible objective space f1 (x ) (minimize) f2 (x ) x 2 (minimize) x 1 feasible decision space Pareto-optimal front B C Pareto-optimal solutions A feasible objective space f1 (x ) (minimize) f2 (x ) x 2 (minimize) x 1 feasible decision space x 2 x 1 feasible decision space Pareto-optimal front B C Pareto-optimal solutions A 9 Goals in MOO ? Find set of solutions as close as possible to Paretooptimal front ? To find a set of solutions as diverse as possible feasible objective space f 1 (x ) f 2 (x ) Pareto-optimal front 1 2 Classic MOO Methods 11 Weighted Sum Method ? Scalarize a set of objectives into a single objective by adding each objective pre-multiplied by a usersupplied weight ? Weight of an objective is chosen in proportion to the relative importance of the objective x x x i n h k K g j J F w f U i i L i k j M m m m , 1, 2, , ( ) 0, 1, 2, , ( ) 0, 1, 2, , ( ), ( ) ( ) 1 L L subject to L minimize ( ) ≤ ≤ = = = ≥ = = ∑ = x x x x 12 ? Advantage ?Simple ? Disadvantage ?It is difficult to set the weight vectors to obtain a Pareto-optimal solution in a desired region in the objective space ?It cannot find certain Pareto-optimal solutions in the case of a nonconvex objective space Weighted Sum Method 13 Weighted Sum Method (Convex Case) f1 f2 w 2 w 1 Paretooptimal front Feasible objective space 14 Weighted Sum Method (Non-Convex Case) f1 f2 Paretooptimal front Feasible objective space 15 ε-Constraint Method ? Haimes et. al. 1971 ? Keep just one of the objective and restricting the rest of the objectives within user-specific values x x x i n h k K g j J f m M m f U i i L i k j m m , 1, 2, , ( ) 0, 1, 2, , ( ) 0, 1, 2, , ( ) , 1, 2, , and ( ), ( ) ( ) L L L subject to L minimize ≤ ≤ = = = ≥ = ≤ = ≠ x x x x ε μ μ 16 ε-Constraint Method Keep f2 as an objective Minimize f2(x) Treat f1 as a constraint f1(x) ≤ ε1 f1 f2 Feasible objective space ε1a ε1b a b f1 f2 Feasible objective space ε1a ε1b a b 17 ? Advantage ?Applicable to either convex or non-convex problems ? Disadvantage ?The ε vector has to be chosen carefully so that it is within the minimum or maximum values of the individual objective function ε-Constraint Method 18 Weighted Metric Method ? Combine multiple objectives using the weighted distance metric of any solution from the ideal solution z * x x x i n h k K g j J l w f z U i i L i k j p M m p m m m , 1, 2, , ( ) 0, 1, 2, , ( ) 0, 1, 2, , ( ) , ( ) ( ) 1/ 1 * L L subject to L minimize ( ) p ≤ ≤ = = = ≥ = ? ? ? ? ? ? = ∑ − = x x x x 19 Weighted Metric Method f1 f2 z * a b p=1 (Weighted sum approach) f1 f2 z * a b p=2 20 Weighted Metric Method f1 f2 z * a b p= ∞ (Weighted Tchebycheff problem) 21 ? Advantage ?Weighted Tchebycheff metric guarantees finding all Pareto-optimal solution with ideal solution z* ? Disadvantage ?Requires knowledge of minimum and maximum objective values ?Requires z* which can be found by independently optimizing each objective functions ?For small p, not all Pareto-optimal solutions are obtained ?As p increases, the problem becomes non-differentiable Weighted Metric Method Multi-Objective Genetic Algorithms 23 Advantages of GAs over Traditional Methods ? Our desired answer: a set of solutions ? Traditional optimization methods operate on a candidate solution ? Genetic algorithms fundamentally operate on a set of candidate solutions 24 Multi-Objective EAs (MOEAs) ? There are several different multi-objective evolutionary algorithms ? Depending on the usage of elitism, there are two types of multi-objective EAs 25 Multi-Objective MOEAs Non-Elitist MOEAs ? Vector evaluated GA ? Vector optimized ES ? Weight based GA ? Random weighted GA ? Multiple-objective GA ? Non-dominated Sorting GA ? Niched Pareto GA Elitist MOEAs ? Elitist Non-dominated Sorting GA ? Distance-based Pareto GA ? Strength Pareto GA ? Pareto-archived ES 26 Identifying the Non-Dominated Set ? Critical Step in Elitist Strategies ? Kung’s et. al. Method ?Computational the most efficient method known ?Recursive 27 Kung’s et. al. Method: Step 1 ? Step 1: Sort population in descending order of importance of the first objective function and name population as P ? Step 2: Call recursive function Front(P) 28 Front(P) IF |P| = 1, Return P ELSE T = Front ( P(1: [ |P|/2 ]) ) B = Front ( P( [ |P|/2 + 1 ] : |P|) ) IF the i-th non-dominated solution of B is not dominated by any nondominated solution of T, M=T ∪{i} Return M END END 29 Notes on Front (P) ? |•| is the number of the elements ? P( a : b ) means all the elements of P from index a to b, ? [ •] is an operator gives the nearest smaller integer value 30 Example of Kung’s Method f1 (min) f2 (min) abc d efg h f1 (min) f2 (min) abc d efg h 31 Example of Kung’s Method a b e c f h d g a b e c f h d g n recursively call the function ‘front’ Step 1 a b e c f h d g Step 2 a a b b e e c c f f h h d d g g a a b b e e c c f f h h d d g g a f, h e d, g a, e, h f, h a, e o front returns M as output T B T T B B B B B B T T T T a, b, e, c, f, h d, g a, b, e, c f, h, d, g a,b e,c f, h d, g 32 Elitist MOEAs ? Elite-preserving operator carries elites of a population to the next generation ? Rudolph(1996) proved GAs converge to the global optimal solution of some functions in the presence of elitism ? Elitist MOEAs two methods are often used ? Elitist Non-dominated Sorting GA (NSGA II) ? Strength Pareto EA * Reference: G. Rudolph, Convergence of evolutionary algorithms in general search spaces, In Proceedings of the Third IEEE conference of Evolutionary Computation, 1996, p.50-54 33 Elitist Non-Dominated Sorting GA (Deb et al., 2000) ? Use an explicit diversity-preserving strategy together with an elite-preservation strategy 34 Elitist Non-Dominated Sorting GA ? Non-Dominated Sorting ?Classify the solutions into a number of mutually exclusive equivalent non-dominated sets U ρ =1 = j F F j F2 F1 F3 (min) f1 (min) f2 35 Elitist Non-Dominated Sorting GA ? Determine Crowding Distance ?Denotes half of the perimeter of the enclosing cuboid with the nearest neighboring solutions in the same front Cuboid (min) f2 (min ) f1 i 36 Elitist Non-Dominated Sorting GA ? Crowding tournament selection ? Assume that every solution has a non-domination rank and a local crowding distance ? A solution i wins a tournament with another solution j 1. if the solution i has a better rank 2. They have the same rank but solution i has a better crowing distance than solution j 37 Elitist Non-Dominated Sorting GA ? Step 1 ?Combine parent P t and offspring Q t populations Rt = P t ∪ Q t ?Perform a non-dominated sorting to Rt and find different fronts Fi ? Step 2 ?Set new population Pt+1 = ∅ and set i = 1 ?Until |Pt+1| + |Fi| < N, perform Pt+1 = Pt+1 ∪ Fi and increase i 38 Elitist Non-dominated Sorting GA ? Step 3 ?Include the most widely spread solutions (N-|Pt+1|) of Fi in Pt+1 using the crowding distance values ? Step 4 ?Create offspring population Qt+1 from Pt+1 by using the crowded tournament selection, crossover and mutation operators 39 Elitist Non-Dominated Sorting GA Pt Qt Rt Non-dominated sorting Pt+1 Crowding distance sorting F1 F2 F3 F1 F2 F3 discard 40 Elitist Non-dominated Sorting GA ? Advantages ? The diversity among non-dominated solutions is maintained using the crowding procedure: No extra diversity control is needed ? Elitism protects an already found Pareto-optimal solution from being deleted 41 Elitist Non-dominated Sorting GA ? Disadvantage ? When there are more than N members in the first nondominated set, some Pareto-optimal solutions may give their places to other non-Pareto-optimal solutions N=7 A Pareto-optimal solution is discarded 42 Strength Pareto EA (SPEA) ? Zitzler & Thiele., 1998 ? Use external population P ? Store fixed number of non-dominated solutions ? Newly found non-dominated solutions are compared with the existing external population and the resulting non-dominated solutions are preserved 43 SPEA Clustering Algorithm 1. Initially, each solution belongs to a distinct cluster Ci 2. If number of clusters is less than or equal to N, go to 5 3. For each pair of clusters, calculate the cluster distance dij and find the pair with minimum cluster-distance 4. Merge the two clusters and go to 2 5. Choose only one solution from each cluster and remove the other (The solution having minimum average distance from other solutions in the cluster can be chosen) ∑ ∈ ∈ = 1 2 1 2 , 12 ( , ) 1 i C j C d i j C C d 44 SPEA Clustering Algorithm f 2 f 1 Cluster 1 Cluster 2 Cluster 3 N=3 45 SPEA Algorithm ? Step 1. Create initial population P 0 of size N randomly and an empty external population P0 with maximum capacity of N ? Step 2. Find the non-dominated solutions of P t and copy (add) these to P t ? Step 3. Find the non-dominated solutions of P t and delete all dominated solutions ? Step 4. If |P t| > N then use the clustering technique to reduce the size to N 46 ? Step 5 Fitness evaluation ? Elites: assign fitness to each elite solution i by using ? For current population: assign fitness to a current population member j where Dj is the set of all external population members dominating j ? Note: a solution with smaller fitness is better ∑ ∈ = + Dj i Fj 1 Si N 1 # of population members dominated by elite + = i Si 47 N 1 # of current population members dominated by Sa + = a 7 2 = ∑ ∈ = + Dj i i F 1 S 1 7 9 7 2 =1+ = SPEA Algorithm Fitness Eval. f2 f1 P P a 1 2 3 4 5 6 2/7 2/7 3/7 9/7 14/7 10/7 10/7 9/7 7/7 N= 6, N=3 48 SPEA Algorithm ? Step 6 ?Apply a binary tournament selection (in a minimization sense), a crossover and a mutation operator to Pt U Pt and generate the next population Pt+1 49 Advantages of SPEA ? Once a solution Pareto-optimal front is found, it is stored in the external population ? Clustering ensures a spread among the nondominated solutions ? The proposed clustering algorithm is parameterless 50 Disadvantages of SPEA ? A balance between the regular population size N and the external population size N is important ?If N is too large, selection pressure for the elites is too large and the algorithm may not converge to the Pareto-optimal front ?If N is too small, the effect of elitism will be lost