Comparative Analysis of Genetic Algorithm and Classical Algorithms in Fractional Programming

Abstract

This paper compares the performances of genetic algorithm with various classical algorithms in solving fractional programming. Genetic algorithm is one of the new forms of algorithms for solving optimization problems, which may not be efficient but a generic way to solve nonlinear optimization problems. The traditional optimization algorithms have difficulty in computing the derivatives and second order partial derivatives, i.e., Hessian for the fractional function. The issues of discontinuity seriously affects traditional algorithm. There are large numbers of classical methods for searching the optimum point of nonlinear functions. The classical search algorithms may be largely classified as gradient based methods and nongradient methods. Here, a comparative performance analysis of different algorithms is made through a newly defined function called algorithmic index. An algorithm based on heuristics for computation of gewicht vector required to derive algorithmic index has also been proposed here.

Appendix

Algorithm for Random Search

• Step 1: Choose initial x⁰, z⁰, ϵ such that the minimum lies in (x⁰ − 1/2z⁰, x⁰ + 1/2z⁰). For each Q block, set q = 1 and p = 1.

• Step 2: For i = 1,2…N, create points using uniform distribution of m in the range (–0.5,0.5). Set x ^(p)_i  = x ^q−1_i  + mz ^q−1_i .

• Step 3: If x^(p) is infeasible and p < P, repeat Step 2. If x^(p) is feasible, save x^(p) and f(x^(p)). Increment p and repeat step 2;

  Else if p = P, set x^q to be the point that has lowest f(x^(p)) over all feasible x^(p) including x^q−1

  And reset p = 1.

• Step 4: Reduce the range via z ^q_i  =  ϵ z ^q−1_i .

• Step 5: If q > Q. Stop.

  Else increment q and continue to Step 2.

Box’s Evolutionary Algorithm

• Step 1: Choose initial point. Choose size reduction step δ_i and termination criteria ϵ _..

• Step 2: If δ_i <  ϵ _. STOP.

• Step 3: Else create 2^N points by adding and subtracting δ_i from each variable at the initial point.

• Step 4: Compute function values at all 2^N points. Find the optimum among these points. Set it as initial point for next iteration.

• Step 5: Reduce size of the step to δ_i/2 and go to Step 2.

Hooke’s Jeeves’ Algorithm

• Step 1: Initial point is selected and objective function is evaluated.

• Step 2: Search is made in the direction of each dimension by a step size S_i to find lowest of functional value.

• Step 3: In case the function value does not decrease in any direction, the step size is reduced and fresh search is made.

• Step 4: If the value of objective function reduces, a new initial point is found as follows:

  X ^(k+1)_i,o  = X ^k+1_i + θ (X ^k+1_i  − X ^k_i ), θ  > 1.

• Step 5: This search continues till the termination criteria is met, i.e., θ  <  ϵ .

Gradient Descent Method

• Step 1: Choose initial point x⁽⁰⁾ and termination parameters ϵ ₁ and ϵ _2.

• Step 2: Compute first derivative $\nabla {\text{f}}\left( {x^{k} } \right).$

• Step 3: If $\left| {\left| {\nabla {\text{f}}\left( {x^{k} } \right)} \right|} \right| \le$ ϵ ₁ STOP.

  Else go to next step

• Step 4: By unidirectional search, find α ^k such that f(x^(k+1)) = f(x^k −  α ^k $\nabla {\text{f}}\left( {x^{k} } \right)$) is minimum. One criteria for termination is | $\nabla {\text{f}}\left( {x^{k + 1} } \right).\nabla {\text{f}}\left( {x^{k} } \right)$ |  ≤   ϵ _2.

• Step 5: If $\frac{{||x^{k + 1} - x^{k} ||}}{{\left| {\left| {x^{k}} \right|} \right|}} \le$ ϵ ₁, then STOP.

  Else set k = k + 1, go to step 2.

Genetic Algorithm

• Start and generate a random population of size n.

• Fitness: Evaluate fitness of each chromosome.

• New Population: Create new population by repeating the steps below

• Select two parent chromosomes from the population according to best fitness.

• Cross over the parents, with a crossover probability to form new population.

• With a mutation probability, mutate the new offspring.

• Add the new offspring in the population.

• Replace: Use the new generation for next iteration.

• Test: Check termination criteria.

• Loop: Go to step 2.

MATLAB Code for Estimating Gewicht Vector