Search results
Results from the WOW.Com Content Network
A step of the Frank–Wolfe algorithm Initialization: Let , and let be any point in . Step 1. Direction-finding subproblem: Find solving Minimize () Subject to (Interpretation: Minimize the linear approximation of the problem given by the first-order Taylor approximation of around constrained to stay within .)
An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. [1] The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, [2] which runs in provably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...
For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context is the marginal cost of the ...
f : ℝ n → ℝ is the objective function to be minimized over the n-variable vector x, g i (x) ≤ 0 are called inequality constraints; h j (x) = 0 are called equality constraints, and; m ≥ 0 and p ≥ 0. If m = p = 0, the problem is an unconstrained optimization problem. By convention, the standard form defines a minimization problem.
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.
Consider the following nonlinear optimization problem in standard form: . minimize () subject to (),() =where is the optimization variable chosen from a convex subset of , is the objective or utility function, (=, …,) are the inequality constraint functions and (=, …,) are the equality constraint functions.
When minimizing a function f in the neighborhood of some reference point x 0, Q is set to its Hessian matrix H(f(x 0)) and c is set to its gradient ∇f(x 0). A related programming problem, quadratically constrained quadratic programming , can be posed by adding quadratic constraints on the variables.
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} .