MULTI-OBJECTIVE OPTIMIZATION

Transcription

1 MULTI-OBJECTIVE OPTIMIZATION Introduction Many real-world problems require the simultaneous optimization of a number of objective functions. Some of these objectives may be in conflict. Example 1:optimal routes in data communications networks. Objectives: minimize routing cost, minimize route length, minimize congestion, maximize utilization of physical infrastructure. Example 2:maximizing structural stability vs. minimizing costs of a mechanical structure. 149

2 Multi-objective problem x = (x 1, x 2,, x n ) f(x) = ( f 1 (x), f 2 (x),, f k (x)) Multi-objective problem: minimize f( x) subject to gm( x) 0, m = 1,, ng h ( x) = 0, m = n + 1,, n + n x [ x, x ] m g g h min max 150 What is an optimum? Improving in one objective may deteriorate another. Balance in trade-off solutions is achieved when A solution cannot improve any objective without degrading one or more of the other objectives. These solutions are called non-dominated solutions. The set of these solutions is a non-dominated setor the Pareto-optimal set. The corresponding objective vectors are referred to as the Pareto-front. 151

3 Pareto-optimality Domination:a decision vector x 1 dominates a decision vector x 2 (denoted as x x ), if and only if 1 2 x 1 is not worse than x 2 in all objective, i.e. f k (x 1 ) f k (x 2 ), k. x 1 is strictly better than x 2 in at least one objective, i.e. k: f k (x 1 ) < f k (x 2 ). A similar concept can be defined for objective vectors. Objective vector dominance is denoted by f. 1 2 A decision vector x * is Pareto-optimalif does not exist a vector x x * that dominates it. f 152 Pareto-optimality Pareto-optimal set, P *, is the set of all Pareto-optimal decision vectors. Pareto-optimal front, PF *, contains all non-dominated objective vectors. In practice, it is very difficult to find the Pareto-optimal front. The concepts of ε-dominance and ε-approximation are applied in real problems. 153

4 Example From: Example: truss f 1 (x) minimize volume of truss f 2 (x) minimize joint displacement 155

5 Global Pareto front Coello Coello, Carlos A., A Short Tutorial on Evolutionary Multiobjective Optimization 156 Multi-objective problems Aggregating functions. Most intuitive approach: combine multiple objectives into a single function, e.g. using sum of weights: k i i i= 1 min w f ( x) Advantages: does not require any changes to optimization algorithm with only one objective. Drawbacks:difficult to generate weights. Cannot generate Pareto decisions when Pareto front is concave. 157

6 Ideal multiobjective optimization (Deb) 1. Find multiple trade-off solutions with a wide range of values. 2. Choose one of the obtained solutions using high-level info. 158 Multi-objective GA approaches Vector Evaluated GA (VEGA), (Shaffer, 1985). Multi-Objective GA (MOGA), (Fonseca & Fleming, 1993) Non-dominated Sorting GA (NSGA), (Deb et al., 1994). Niched Pareto GA (NPGA), (Horn et al., 94) Target Vector approaches, (several authors) NSGA II, (Deb et al., 2002). Deb K, PratapA, AgarwalS, et al. A fast and elitist multiobjectivegenetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6 (2): Apr Times Cited: 6307 (Dec. 16, 2014). 159

7 Vector Evaluated GA (VEGA) With Mobjectives to be handled, population is divided by these objectives. Each subpopulation has its own fitness. Coello Coello, Carlos A., A Short Tutorial on Evolutionary Multiobjective Optimization 160 VEGA Advantages: only selection mechanism is modified, so it is easy to implement and efficient (computational complexity is the same). Drawbacks:difficult to find good compromise solutions, as each solution is looking only to an individual objective function. It can happen that only a few points of the Pareto front are found. 161

8 Multi-Objective GA (MOGA) Differs in the way fitness is assigned to a solution. A rankis assigned to each solution r i = 1 + n i, where n i is the number of solutions that dominate solution i. Fitness is related to the inverse of ranking. This simple procedure does notassure diversityamong non-dominated solutions. A niche-formation methodwas introduced to distribute the population over the Pareto-optimal region. 162 MOGA Advantages: fitness assignment scheme is simple. Can find spread Pareto-optimal solutions. Drawbacks: can introduce unwanted bias towards some solutions. May be sensitive to the shape of Pareto-optimal front. 163

9 Non-dominated Sorted GA (NSGA II) 1. Uses an elitist principle. 2. Uses an explicit diversity preserving mechanism. 3. Emphasizes non-dominated solutions in a population. Example(Deb): classification in three non-dominated fronts. 164 NSGAII: Examples (Deb) 165

10 Multi-objective ACO Three classes: 1. Multi-colony algorithms 2. Single colony with multiple pheromone matrices algorithms. 3. Single colony with multiple heuristic functions algorithms. 166 Multi-objective ACO C. Garcia-Martinez, O. Cordon and F. Herrera (2007) 167

11 Multi-colony algorithms Each colony optimize one objective. Having k objectives, a total of kcolonies is used. Colonies cooperate by sharing information about the solutions found by each colony. Local sharing:is performed after next node is added to current path of a new partial solution. Solutions are grouped into non-dominance solutions. Fitness value f ij is calculated for the best solution so far. Global sharing: similar process but now it is performed after completion of paths. 168 Multiple pheromone matrices Two objectives: two pheromone matrices and two heuristic matrices (Iredi, 2001): Having n c objectives (Doerner, 2004): 169

12 Multiple heuristic methods Single pheromone function and several heuristics information functions (Barán and Schaerer, 2003): 170 Example Traveling Salesman Problem with multiple objectives: cost, length, travel time, or tourist attractiveness. Several instances ( number of cities). 171

13 Results: Kroab Results: Kroab