Question 13: In convex optimization, what does it mean for a solution to be a global minimum? A) It is the only solution to the problem B) It is the best solution among all feasible solutions C) It is the solution with the lowest objective value D) It is the solution with the highest objective value
Answer (1)
In the context of convex optimization, a global minimum refers to a solution point in the feasible set of an optimization problem where the objective function takes its smallest possible value over all possible feasible solutions.
To better understand this, let's break this down step-by-step:
Convex Optimization : This is a subfield of optimization that deals with problems where the objective function is convex, and the feasible set is a convex set. A function is convex if any line segment between two points on its graph never lies below the graph itself.
Feasible Solutions : These are the solutions that satisfy all the constraints of the optimization problem. In simpler terms, feasible solutions are those that can actually occur given the problem's requirements.
Global Minimum : A global minimum is a point in the feasible set where the objective function, let's say f ( x ) , has the lowest possible value of f ( x ) over all the feasible solutions. This means that no other feasible solution has a value of the objective function less than this point's value.
Choice B ("It is the best solution among all feasible solutions") is the correct answer here as it accurately describes a global minimum in the context of convex optimization. It emphasizes that the global minimum is the optimal choice among all feasible options.
A global minimum is not necessarily the only solution, because in some cases, there might be multiple solutions that achieve the same lowest objective value. Nonetheless, it guarantees that no other feasible solution has a lower objective value than the global minimum, which is key in optimization problems.
