What is a constraint in an optimization problem?

Constrained optimization problems are problems for which a function is to be minimized or maximized subject to constraints . stands for “maximize subject to constraints “. You say a point satisfies the constraints if is true.

What are convex constraints?

A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. With a convex objective and a convex feasible region, there can be only one optimal solution, which is globally optimal.

What is affine equality constraint?

Equalities defined be affine functions define a linear subspace and therefore a convex set. Equalities defined by convex functions define a manifold which is, in general, NOT a convex set. If h(x) is a convex function then (1) is a convex constraint. However, (2) is not a convex constraint anymore.

What are affine constraints?

A constraint of the form where a is a coefficient vector, b is a real scalar, x is a vector of variables and is one of , or . Most people refer to such constraints as “linear”, but technically a linear function passes through the origin (which in this case would imply ) whereas an affine function need not. 4.6K views.

What are the two types of constraints in constrained optimization?

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.

What is convex set in optimization?

Definition. A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible set is a convex set. A function mapping some subset of into is convex if its domain is convex and for all and all in its domain, the following condition holds: .

Is equality constraint convex?

The objective is convex. The equality constraint function is not affine; however, we can rewrite it as x1 = −x2 which is then a linear equality constraint. The inequality constraint function is not convex; however, we can rewrite it as x1 ≤ 0 which again is linear.

What does affine mean in math?

In geometry, an affine transformation or affine map (from the Latin, affinis, “connected with”) between two vector spaces consists of a linear transformation followed by a translation. In a geometric setting, these are precisely the functions that map straight lines to straight lines.

Is affine function linear?

An affine function is a function composed of a linear function + a constant and its graph is a straight line. The general equation for an affine function in 1D is: y = Ax + c. An affine function demonstrates an affine transformation which is equivalent to a linear transformation followed by a translation.