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Showing content from http://accord-framework.net/docs/html/T_Accord_Math_Optimization_QuadraticObjectiveFunction.htm below:

Quadratic objective function.

Accord.Math (in Accord.Math.dll) Version: 3.8.0

public class QuadraticObjectiveFunction : NonlinearObjectiveFunction, 
	IObjectiveFunction

Public Class QuadraticObjectiveFunction
	Inherits NonlinearObjectiveFunction
	Implements IObjectiveFunction

The QuadraticObjectiveFunction type exposes the following members.

Determines whether the specified object is equal to the current object.

Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.

Serves as the default hash function.

Gets the Type of the current instance.

Creates a shallow copy of the current Object.

Returns a String that represents this instance.

Attempts to create a QuadraticObjectiveFunction from a String representation.

Attempts to create a QuadraticObjectiveFunction from a String representation.

Checks whether an object implements a method with the given name.

Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices.

Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime.

Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime.

In mathematics, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x², y², z², xy, xz, yz, x, y, z, and a constant:

f(x,y,z) = ax² + by² +cz² + dxy + exz + fyz + gx + hy + iz + j

Please note that the function's constructor expects the function expression to be given on this form. Scalar values must be located on the left of the variables, and no term should be duplicated in the quadratic expression. Please take a look on the examples section of this page for some examples of expected functions.

References:

• Wikipedia, The Free Encyclopedia. Quadratic Function. Available on: https://en.wikipedia.org/wiki/Quadratic_function

Examples of valid quadratic functions are:

var f1 = new QuadraticObjectiveFunction("x² + 1");
var f2 = new QuadraticObjectiveFunction("-x*y + y*z");
var f3 = new QuadraticObjectiveFunction("-2x² + xy - y² - 10xz + z²");
var f4 = new QuadraticObjectiveFunction("-2x² + xy - y² + 5y");

It is also possible to specify quadratic functions using lambda expressions. In this case, it is first necessary to create some dummy symbol variables to act as placeholders in the quadratic expressions. Their value is not important, as they will only be used to parse the form of the expression, not its value.

double x = 0, y = 0, z = 0;

var g1 = new QuadraticObjectiveFunction(() => x * x + 1);
var g2 = new QuadraticObjectiveFunction(() => -x * y + y * z);
var g3 = new QuadraticObjectiveFunction(() => -2 * x * x + x * y - y * y - 10 * x * z + z * z);
var g4 = new QuadraticObjectiveFunction(() => -2 * x * x + x * y - y * y + 5 * y);

Finally, for large problems, it is usually best to declare quadratic problems using matrices and vectors. Without loss of generality, the quadratic matrix can always be taken to be symmetric (as the anti- symmetric part has no contribution to the function). For efficiency reasons, the quadratic matrix must be symmetric.

var h1 = new QuadraticObjectiveFunction("3x² - 4y² + 6xy + 3x + 2y");

double[,] Q = { { 6, 6 }, { 6, -8 } }; 
double[] d = { 3, 2 };


var h2 = new QuadraticObjectiveFunction(Q, d);

After those functions are created, you can either query their values using

f1.Function(new [] { 5.0 }); 

Or you can pass it to a quadratic optimization method such as Goldfarb-Idnani to explore its minimum or maximal points:

double x = 0, y = 0, z = 0;


var f = new QuadraticObjectiveFunction(() => x * x - 2 * x * y + 3 * y * y + z * z - 4 * x - 5 * y - z);


var constraints = new List<LinearConstraint>();
constraints.Add(new LinearConstraint(f, () => 6 * x - 7 * y <= 8));
constraints.Add(new LinearConstraint(f, () => 9 * x + 1 * y <= 11));
constraints.Add(new LinearConstraint(f, () => 9 * x - y <= 11));
constraints.Add(new LinearConstraint(f, () => -z - y == 12));


GoldfarbIdnani solver = new GoldfarbIdnani(f, constraints);


bool success = solver.Minimize();

double value = solver.Value;
double[] solutions = solver.Solution;

Sometimes it is easiest to compose quadratic functions as linear combinations of other quadratic functions. The QuadraticObjectiveFunction supports this by overloading the addition and multiplication operators.

var f1 = new QuadraticObjectiveFunction("2x² + 4y² - 2xy + 6");
var f2 = new QuadraticObjectiveFunction("3x² - 4y² + 6xy + 3x + 2y");










QuadraticObjectiveFunction f = f1 + (2 * f2); 


double[] x = { 1, 2 };

double result1 = f1.Function(x);
double result2 = f2.Function(x);

double result = f.Function(x);          
double check = result1 + 2 * result2;   

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