Search results
Results from the WOW.Com Content Network
Constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints, which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied.
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 ...
Minimize subject to the algebraic constraints = () Depending upon the type of direct method employed, the size of the nonlinear optimization problem can be quite small (e.g., as in a direct shooting or quasilinearization method), moderate (e.g. pseudospectral optimal control [ 11 ] ) or may be quite large (e.g., a direct collocation method [ 12
The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization.Also known as the conditional gradient method, [1] reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956. [2]
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.
In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank.
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. A maximization problem can be treated by negating the objective function.
This constraint is written in standard form by defining a new penalty function y(t) = a(t) − b(t). The above problem seeks to minimize the time average of an abstract penalty function p'(t)'. This can be used to maximize the time average of some desirable reward function r(t) by defining p(t) = −r('t).