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The idea is to substitute the constraint into the objective function to create a composite function that incorporates the effect of the constraint. For example, assume the objective is to maximize f ( x , y ) = x ⋅ y {\displaystyle f(x,y)=x\cdot y} subject to x + y = 10 {\displaystyle x+y=10} .
The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
For any greater-than constraints, introduce surplus s i and artificial variables a i (as shown below). Choose a large positive Value M and introduce a term in the objective of the form −M multiplying the artificial variables. For less-than or equal constraints, introduce slack variables s i so that all constraints are equalities.
The necessary conditions are sufficient for optimality if the objective function of a maximization problem is a differentiable concave function, the inequality constraints are differentiable convex functions, the equality constraints are affine functions, and Slater's condition holds. [11]
The function f is variously called an objective function, criterion function, loss function, cost function (minimization), [8] utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes) the objective function is called an optimal solution.
The equality constraint functions :, =, …,, are affine transformations, that is, of the form: () =, where is a vector and is a scalar. The feasible set C {\displaystyle C} of the optimization problem consists of all points x ∈ D {\displaystyle \mathbf {x} \in {\mathcal {D}}} satisfying the inequality and the equality constraints.
In this way, all lower bound constraints may be changed to non-negativity restrictions. Second, for each remaining inequality constraint, a new variable, called a slack variable, is introduced to change the constraint to an equality constraint. This variable represents the difference between the two sides of the inequality and is assumed to be ...
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.