Convex optimal value

  • Is optimal set convex?

    every local minimum is a global minimum; the optimal set is convex; if the objective function is strictly convex, then the problem has at most one optimal point..

  • What does convex valued mean?

    In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points.
    Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set..

  • Consider the function f(x) = x2, which is a strictly convex function.
    The unique global minimum of this function in R is x = 0.
Mar 3, 2022For a μ strong convex and L smooth function f:Rn→R, assume its unique global optimal point is x∗. Intuitively, if our current x is very close  Finding the optimal value in an optimization problemConvex optimization - comparing optimal values (any counter Find optimal set and optimal value for feasible setFinite optimal value for a linear program with unbounded feasible More results from math.stackexchange.com
Mar 3, 2022For a μ strong convex and L smooth function f:Rn→R, assume its unique global optimal point is x∗. Intuitively, if our current x is very close  Finding the optimal value in an optimization problemConvex optimization - comparing optimal values (any counter How to show that a certain value function is convex?Find optimal set and optimal value for feasible setMore results from math.stackexchange.com

Do all convex optimization problems have the same optimal value?

In practice, nearly all convex problems satisfy some type of constraint qualification, and hence the primal and dual problem have the same optimal value

For an unconstrained convex optimization problem, we know we are at the global minimum if the gradient is zero

Generally speaking, the theory of Lagrange duality is the study of optimal solutions to convex optimization problems. As we saw previously in lecture, when minimizing a differentiable convex function f(x) with respect to x ∈ Rn, a necessary and sufficient condition for x∗ ∈ Rn to be globally optimal is that ∇xf(x∗) = 0.
Convex optimal value
Convex optimal value

Function whose values are sets (mathematics)

A set-valued function is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set.
Set-valued functions are used in a variety of mathematical fields, including optimization, control theory and game theory.

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