Certain convex minimization problem

In summary, the conversation discusses the use of inequality signs in an optimization problem, specifically in regards to the standard form and the use of the Lagrangian method and KKT conditions. It is noted that the inequality sign does not make a difference in the solution, as long as the constraint is defined appropriately. The speaker also mentions that the 'normal' KKT conditions can be obtained by defining the appropriate function.
  • #1
nikozm
54
0
Hi, I would like to know if the inequality sign plays any role to the following optimization problem:

minimize f0(x)
subject to f1(x)>=0

where both f0(x) and f1(x) are convex. The standard form of these problems require a constraint such as: f1(x)<=0, but i am interested in the opposite condition (f1(x)>=0) and solving it using the standard Lagrangian method and KKT conditions. Is this right?

Any help would be useful. Thanks in advance!
 
  • #3
The method essentially says that the solution occurs when the inequality is strict, and you can just find points where the gradient is zero, or the inequality is an equality, and you can use Lagrangian multipliers. It doesn't matter whether the inequality is less than or greater than, everything I said above still holds.
 
  • #4
f1(x)>=0 is the same as -f1(x)<=0, so just define f1 appropriately and you get the 'normal' KKT conditions.
 

1. What is a convex minimization problem?

A convex minimization problem refers to a type of optimization problem where the goal is to find the minimum value of a convex function subject to certain constraints. A convex function is one that is always curved upwards, and the constraints are conditions that must be satisfied by the variables in the problem.

2. How is a convex minimization problem solved?

A convex minimization problem can be solved using various algorithms, such as gradient descent or interior point methods. These algorithms involve iteratively updating the variables until the minimum value is reached. Additionally, certain convex minimization problems may have closed-form solutions that can be found using mathematical techniques.

3. What are some real-world applications of convex minimization problems?

Convex minimization problems have a wide range of applications in fields such as engineering, economics, and machine learning. For example, they can be used to optimize the design of structures, find the most efficient resource allocation in a company, or train a machine learning model to make accurate predictions.

4. How do you determine if a problem is a convex minimization problem?

A problem can be determined to be a convex minimization problem if the objective function and constraints meet certain criteria. The objective function must be convex, and the constraints must be convex as well. This means that the function and constraints must satisfy specific mathematical properties, such as being always curved upwards.

5. Can convex minimization problems have non-unique solutions?

Yes, it is possible for convex minimization problems to have multiple solutions that meet the minimum value. This can happen when the objective function has a flat region, and any point within that region will have the same minimum value. In such cases, any of these points can be considered a valid solution to the problem.

Similar threads

A Finding dual optimum of a linear problem
    • General Math
Replies
0
Views
704
MHB How to solve following optimization problem?
    • General Math
Replies
2
Views
947
Optimization: How to find the dual problem?
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Optimization: Dual for L1 norm minimization with equality constraint
    • Calculus and Beyond Homework Help
Replies
1
Views
563
MHB Convex function and convex set
    • Calculus
Replies
3
Views
3K
MHB What are some recommended books for optimisation?
    • Calculus
Replies
2
Views
2K
Convex Optimization: Dual Function Definition
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
A Solving polynomial coefficients to minimize square error
    • Calculus
Replies
8
Views
2K
A Finding Minimal Mean Distance Curves on the Unit Sphere
    • General Math
Replies
2
Views
1K
I Best way to fit three functions
    • General Math
Replies
2
Views
728

Hot Threads

Recent Insights

Back
Top