The following program calculates the minimum point of a multi-variable function using Newton's method. This method is implemented in a basic form without any enhancements.
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 values for the initial set of variables
2. The values for the tolerances for each variable.
3. The function tolerance
4. The maximum number of iterations
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 final results:
1. The coordinates of the minimum value.
2. The fine step sizes for each variable.
3. The minimum function value.
4. The number of iterations
The current code finds the minimum for the following function:
f(x1,x2) = x1 - x2 + 2 * x1 ^ 2 + 2 * x1 * x2 + x2 ^ 2
Using, for each variable, an initial value of 0, initial step size of 0.1, minimum step size of 1e-7, and using a function tolerance of 1e-7. Here is the sample console screen:
Here is the listing for 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_Newton1
{
sealed class Module1
{
static public void Main()
{
int nNumVars = 2;
double[] fX = new double[] { 0, 0 };
double[] fParam = new double[] { 0, 0 };
double[] fToler = new double[] { 0.00001, 0.00001 };
int nIter = 0;
int nMaxIter = 100;
double fEpsFx = 0.0000001;
int i;
double fBestF;
string sAnswer;
string sErrorMsg = "";
COptimNewton1 oOpt;
MyFxDelegate MyFx = new MyFxDelegate(Fx3);
SayFxDelegate SayFx = new SayFxDelegate(SayFx3);
oOpt = new COptimNewton1();
Console.WriteLine("Newton\'s Method for Optimization (basic version)");
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("Tolerance({0}) = {1}", i + 1, fToler[i]);
}
Console.WriteLine("Function tolerance = {0}", fEpsFx);
Console.WriteLine("Maxumum cycles = {0}", nMaxIter);
}
else
{
for (i = 0; i < nNumVars; i++)
{
fX[i] = GetIndexedDblInput("X", i + 1, fX[i]);
fToler[i] = GetIndexedDblInput("Tolerance", i + 1, fToler[i]);
}
fEpsFx = GetDblInput("Function tolerance", fEpsFx);
nMaxIter = GetIntInput("Maxumum cycles", nMaxIter);
}
Console.WriteLine("******** FINAL RESULTS *************");
fBestF = oOpt.CalcOptim(nNumVars, ref fX, ref fParam, ref fToler, fEpsFx, nMaxIter, ref nIter, ref sErrorMsg, MyFx);
if (sErrorMsg.Length > 0)
{
Console.WriteLine("** NOTE: {0} ***", sErrorMsg);
}
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.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 int GetIndexedIntInput(string sPrompt, int nIndex, int nDefInput)
{
string sInput;
Console.Write("{0}({1})? ({2}): ", sPrompt, nIndex, nDefInput);
sInput = Console.ReadLine();
if (sInput.Trim(null).Length > 0)
{
return int.Parse(sInput);
}
else
{
return nDefInput;
}
}
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_Newton1
{
public delegate double MyFxDelegate(int nNumVars, ref double[] fX, ref double[] fParam);
public delegate string SayFxDelegate();
public class COptimNewton1
{
MyFxDelegate m_MyFx;
protected double MyFxEx(int nNumVars, ref double[] fX, ref double[] fParam, ref double[] fDeltaX, double fLambda)
{
int i;
double[] fXX = new double[nNumVars];
for (i = 0; i < nNumVars; i++)
{
fXX[i] = fX[i] + fLambda * fDeltaX[i];
}
return m_MyFx(nNumVars, ref fXX, ref fParam);
}
protected double FirstDeriv(int nNumVars, ref double[] fX, ref double[] fParam, int nIdxI)
{
double fXt, h, Fp, Fm;
fXt = fX[nIdxI];
h = 0.01 *(1 + Math.Abs(fXt));
fX[nIdxI] = fXt + h;
Fp = m_MyFx(nNumVars, ref fX, ref fParam);
fX[nIdxI] = fXt - h;
Fm = m_MyFx(nNumVars, ref fX, ref fParam);
fX[nIdxI] = fXt;
return (Fp - Fm) / 2 / h;
}
protected double SecondDeriv(int nNumVars, ref double[] fX, ref double[] fParam, int nIdxI, int nIdxJ)
{
double fXt, fYt, fHX, fHY, F0, Fp;
double Fm, Fpp, Fmm, Fpm, Fmp, fResult;
// calculate second derivative?
if (nIdxI == nIdxJ)
{
F0 = m_MyFx(nNumVars, ref fX, ref fParam);
fXt = fX[nIdxI];
fHX = 0.01 *(1 + Math.Abs(fXt));
fX[nIdxI] = fXt + fHX;
Fp = m_MyFx(nNumVars, ref fX, ref fParam);
fX[nIdxI] = fXt - fHX;
Fm = m_MyFx(nNumVars, ref fX, ref fParam);
fX[nIdxI] = fXt;
fResult = (Fp - 2 * F0 + Fm) /Math.Pow(fHX, 2);
}
else
{
fXt = fX[nIdxI];
fYt = fX[nIdxJ];
fHX = 0.01 *(1 + Math.Abs(fXt));
fHY = 0.01 *(1 + Math.Abs(fYt));
// calculate Fpp
fX[nIdxI] = fXt + fHX;
fX[nIdxJ] = fYt + fHY;
Fpp = m_MyFx(nNumVars, ref fX, ref fParam);
// calculate Fmm
fX[nIdxI] = fXt - fHX;
fX[nIdxJ] = fYt - fHY;
Fmm = m_MyFx(nNumVars, ref fX, ref fParam);
// calculate Fpm
fX[nIdxI] = fXt + fHX;
fX[nIdxJ] = fYt - fHY;
Fpm = m_MyFx(nNumVars, ref fX, ref fParam);
// calculate Fmp
fX[nIdxI] = fXt - fHX;
fX[nIdxJ] = fYt + fHY;
Fmp = m_MyFx(nNumVars, ref fX, ref fParam);
fX[nIdxI] = fXt;
fX[nIdxJ] = fYt;
fResult = (Fpp - Fmp - Fpm + Fmm) /(4 * fHX * fHY);
}
return fResult;
}
protected void GetFirstDerives(int nNumVars, ref double[] fX, ref double[] fParam, ref double[] fFirstDerivX)
{
int i;
for (i = 0; i < nNumVars; i++)
{
fFirstDerivX[i] = FirstDeriv(nNumVars, ref fX, ref fParam, i);
}
}
protected void GetSecondDerives(int nNumVars, ref double[] fX, ref double[] fParam, ref double[,] fSecondDerivX)
{
int i, j;
for (i = 0; i < nNumVars; i++)
{
for (j = 0; j < nNumVars; j++)
{
fSecondDerivX[i, j] = SecondDeriv(nNumVars, ref fX, ref fParam, i, j);
}
}
}
public double CalcOptim(int nNumVars, ref double[] fX, ref double[] fParam, ref double[] fToler, double fEpsFx, int nMaxIter, ref int nIter, ref string sErrorMsg, MyFxDelegate MyFx)
{
int i;
double fNorm;
double[] g = new double[nNumVars + 1];
int[] Index = new int[nNumVars + 1];
double[] DeltaX = new double[nNumVars + 1];
double[,] j = new double[nNumVars + 1, nNumVars + 1];
bool bStop;
m_MyFx = MyFx;
nIter = 1;
do
{
nIter++;
if (nIter > nMaxIter)
{
sErrorMsg = "Reached maximum iterations limit";
break;
}
GetFirstDerives(nNumVars, ref fX, ref fParam, ref g);
// test if gradient is shallow enough
fNorm = MatrixLibVb.VectNorm(ref g, 2);
if (fNorm < fEpsFx)
{
break;
}
GetSecondDerives(nNumVars, ref fX, ref fParam, ref j);
MatrixLibVb.MV_LUDecomp(ref j, ref Index, nNumVars);
MatrixLibVb.MV_LUBackSubst(ref j, ref Index, nNumVars, ref g);
for (i = 0; i < nNumVars; i++)
{
DeltaX[i] = g[i];
fX[i] -= DeltaX[i];
}
bStop = true;
for (i = 0; i < nNumVars; i++)
{
if (Math.Abs(DeltaX[i]) > fToler[i])
{
bStop = false;
break;
}
}
} while (!bStop);
return MyFx(nNumVars, ref fX, ref fParam);
}
}
}
Copyright (c) Namir Shammas. All rights reserved.