The following program calculates the minimum point of a multi-variable function using random search method. This method starts with an initial point and a search radius for each variable. The algorithm searches for a better optimum point within the search radius. This approach allows the search to drift towards the optimum point without being confined to absolute search ranges.
Click here to download a ZIP file containing the project files for this program.
The program prompts you to either use the predefined default input values or to enter the following:
1. The initial value for each variable:.
2. The search radius for each variable:.
3. The maximum number of iterations per cycle.
4. The function tolerance.
In case you choose the default input values, the program displays these values and proceeds to find the optimum point. In the case you select being prompted, the program displays the name of each input variable along with its default value. You can then either enter a new value or simply press Enter to use the default value. This approach allows you to quickly and efficiently change only a few input values if you so desire.
The program displays the following results:
1. The coordinates of the minimum value.
2. The minimum function value.
3. The number of iterations
Here is a sample session to find the minimum of function:
f(x) = x1 - x2 + 2 * x1 ^ 2 + 2 * x1 * x2 + x2 ^ 2
Using the initial value of 0, range of (-5, 5) for each variable, and using a maximum number of 1000000 iterations and a function tolerance of 1e-7. Here is the sample console screen:
Here is the listing showing the main module. The module contains several test functions:
using System.Diagnostics;
using System.Data;
using System.Collections;
using Microsoft.VisualBasic;
using System.Collections.Generic;
using System;
namespace Optim_RandomSearch3
{
sealed class Module1
{
static public void Main()
{
int nNumVars = 2;
double[] fX = new double[] { 0, 0 };
double[] fParam = new double[] { 0, 0 };
double[] fRadius = new double[] { 2, 2 };
double fEpsFx = 0.0000001;
int nIter = 0;
int nMaxIter = 1000000;
int i;
double fBestF;
string sAnswer;
CRandomSearch3 oOpt;
MyFxDelegate MyFx = new MyFxDelegate(Fx3);
SayFxDelegate SayFx = new SayFxDelegate(SayFx3);
oOpt = new CRandomSearch3();
Console.WriteLine("Random Search (drift scheme) Optimization");
Console.WriteLine("Finding the minimum of function:");
Console.WriteLine(SayFx());
Console.Write("Use default input values? (Y/N) ");
sAnswer = Console.ReadLine();
if (sAnswer.ToUpper() == "Y")
{
for (i = 0; i < nNumVars; i++)
{
Console.WriteLine("X({0}) = {1}", i + 1, fX[i]);
Console.WriteLine("Radius({0}) = {1}", i + 1, fRadius[i]);
}
Console.WriteLine("Maximum iterations = {0}", nMaxIter);
Console.WriteLine("Function tolerance = {0}", fEpsFx);
}
else
{
for (i = 0; i < nNumVars; i++)
{
fX[i] = GetIndexedDblInput("X", i + 1, fX[i]);
fRadius[i] = GetIndexedDblInput("Radius", i + 1, fRadius[i]);
}
nMaxIter = GetIntInput("Maximum iterations", nMaxIter);
Console.Write("Function tolerance? ", null);
fEpsFx = GetDblInput("Function tolerance", fEpsFx);
}
Console.WriteLine("******** FINAL RESULTS *************");
fBestF = oOpt.CalcOptim(nNumVars, ref fX, ref fParam, ref fRadius, nMaxIter, fEpsFx, ref nIter, MyFx);
Console.WriteLine("Optimum at");
for (i = 0; i < nNumVars; i++)
{
Console.WriteLine("X({0}) = {1}", i + 1, fX[i]);
}
Console.WriteLine("Function value = {0}", fBestF);
Console.WriteLine("Number of iterations = {0}", nIter);
Console.WriteLine();
Console.Write("Press Enter to end the program");
Console.ReadLine();
}
static public double GetDblInput(string sPrompt, double fDefInput)
{
string sInput;
Console.Write("{0}? ({1}): ", sPrompt, fDefInput);
sInput = Console.ReadLine();
if (sInput.Trim(null).Length > 0)
{
return double.Parse(sInput);
}
else
{
return fDefInput;
}
}
static public int GetIntInput(string sPrompt, int nDefInput)
{
string sInput;
Console.Write("{0}? ({1}): ", sPrompt, nDefInput);
sInput = Console.ReadLine();
if (sInput.Trim(null).Length > 0)
{
return (int) double.Parse(sInput);
}
else
{
return nDefInput;
}
}
static public double GetIndexedDblInput(string sPrompt, int nIndex, double fDefInput)
{
string sInput;
Console.Write("{0}({1})? ({2}): ", sPrompt, nIndex, fDefInput);
sInput = Console.ReadLine();
if (sInput.Trim(null).Length > 0)
{
return double.Parse(sInput);
}
else
{
return fDefInput;
}
}
static public string SayFx1()
{
return "F(X) = 10 + (X(1) - 2) ^ 2 + (X(2) + 5) ^ 2";
}
static public double Fx1(int N, ref double[] X, ref double[] fParam)
{
return 10 + Math.Pow(X[0] - 2, 2) + Math.Pow(X[1] + 5, 2);
}
static public string SayFx2()
{
return "F(X) = 100 * (X(1) - X(2) ^ 2) ^ 2 + (X(2) - 1) ^ 2";
}
static public double Fx2(int N, ref double[] X, ref double[] fParam)
{
return Math.Pow(100 * (X[0] - X[1] * X[1]), 2) + Math.Pow((X[1] - 1), 2);
}
static public string SayFx3()
{
return "F(X) = X(1) - X(2) + 2 * X(1) ^ 2 + 2 * X(1) * X(2) + X(2) ^ 2";
}
static public double Fx3(int N, ref double[] X, ref double[] fParam)
{
return X[0] - X[1] + 2 * X[0] * X[0] + 2 * X[0] * X[1] + X[1] * X[1];
}
}
}
Notice that the user-defined functions have accompanying helper functions to display the mathematical expression of the function being optimized. For example, function Fx1 has the helper function SayFx1 to list the function optimized in Fx1. Please observe the following rules::
The program uses the following class to optimize the objective function:
using System.Diagnostics;
using System.Data;
using System.Collections;
using Microsoft.VisualBasic;
using System.Collections.Generic;
using System;
namespace Optim_RandomSearch3
{
public delegate double MyFxDelegate(int nNumVars, ref double[] fX, ref double[] fParam);
public delegate string SayFxDelegate();
public class CRandomSearch3
{
public double CalcOptim(int nNumVars, ref double[] fX, ref double[] fParam, ref double[] fRadius, int nMaxIter, double EpsFx, ref int nIter, MyFxDelegate MyFx)
{
double F, fBestF, fLastBestF;
double[] fBestX;
int i;
Random objRand = new Random();
fBestX = new double[nNumVars + 1];
for (i = 0; i < nNumVars; i++)
{
fBestX[i] = fX[i];
}
// calculate and display function value at initial point
fBestF = MyFx(nNumVars, ref fBestX, ref fParam);
if (fBestF > 0)
{
fLastBestF = fBestF + 100;
}
else
{
fLastBestF = 100 - fBestF;
}
nIter = 0;
do
{
nIter++;
if (nIter > nMaxIter)
{
break;
}
for (i = 0; i < nNumVars; i++)
{
fX[i] = fBestX[i] +(objRand.NextDouble() - 0.5) * fRadius[i];
}
F = MyFx(nNumVars, ref fX, ref fParam);
if (F < fBestF)
{
for (i = 0; i < nNumVars; i++)
{
fBestX[i] = fX[i];
}
fBestF = F;
// test function value convergence
if (Math.Abs(fBestF - fLastBestF) < EpsFx)
{
break;
}
fLastBestF = fBestF;
}
} while (true);
return fBestF;
}
}
}
Copyright (c) Namir Shammas. All rights reserved.