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 .)
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 ...
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 (,) = subject to + =.
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.
minimize f(x) subject to x ≤ b. where b is some constant. If one wishes to remove the inequality constraint, the problem can be reformulated as minimize f(x) + c(x), where c(x) = ∞ if x > b, and zero otherwise. This problem is equivalent to the first.
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality.
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.
Consider a family of convex optimization problems of the form: minimize f(x) s.t. x is in G, where f is a convex function and G is a convex set (a subset of an Euclidean space R n). Each problem p in the family is represented by a data-vector Data( p ), e.g., the real-valued coefficients in matrices and vectors representing the function f and ...