Evaluating differential evolution with penalty function to solve constrained engineering problems

A Dynamic Quality-Based Harmony Search Algorithm for Solving Constrained Engineering Optimization Problems

“…The constraints shrink the feasible search-space and, instead of making the problem easier, it makes it more difficult because feasible solutions can be hard to find [1]. Numerical methods, such as gradient-based search, may become difficult and unstable when the objective function and constraints have multiple or sharp peaks [2].…”

“…These meta-heuristics deploy stochastic techniques such as differential evolution (DE) [1,4], particle swarm optimization (PSO) [5,6] and harmony search (HS) [2,7]. Of particular interest is the recent HS approach.…”

“…In this way, the unfeasible solutions can be removed from the population in the next iteration of the algorithm. Using this approach, the constrained problem is treated as an unconstrained one [1,2,4,6,7]. Equation (2) describes a common penalty function used for minimization problems.…”

“…Equation (2) describes a common penalty function used for minimization problems. In equation (2) F(x) represents penalty function, f(x) is the objective function, h(x) (CSE 2013) and g(x) are the equality and inequality constraints as described in (1). Notice that since it is hard to find feasible solutions that satisfy the equality constraint h i (x) = 0 exactly, it is necessary to convert it into the inequality constraint |h i (x)| − ε ≤ 0.…”

“…The DE technique, originally introduced in 1995 [9], was adopted for solving constrained engineering problems [1,4]. DE is a floating-point encoding populational meta-heuristic similar to traditional EAs.…”