Convex optimization dual problem

  • What is the dual form of the optimization problem?

    In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem.
    If the primal is a minimization problem then the dual is a maximization problem (and vice versa)..

  • What is the dual problem in convex optimization?

    Although the primal problem is not required to be convex, the dual problem is always convex. maximization problem, which is a convex optimization problem.
    The Lagrangian dual problem yields a lower bound for the primal problem.
    It always holds true that f⋆ ≥ g⋆, called as weak duality..

  • Why is duality important in convex optimization?

    Optimization problems can be transformed to their dual problems, called Lagrange dual problems, which help to solve the main problem.
    First, with the dual problem one can determine lower bounds for the optimal value of the original problem.
    Second, under certain conditions, the so- lutions of both problems are equal..

  • If the primal problem is a maximization problem, then the dual problem is a minimization problem and vice versa. 3.
    If the primal problem has greater than or equal to type constraints, then the dual problem has less than or equal to type constraints and vice versa.
  • The dual function is concave even when the optimization problem is not convex, since the dual function is the pointwise infimum of a family of affine functions of (λ, ν) (a different affine function for each x ∈ D).
Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact  Dual problemLinear caseNonlinear case

Overview

In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two pers…

Linear case

Linear programming problems are optimization problems in which the objective function and the constraints are all linear. In the primal problem, the objecti…

Nonlinear case

In nonlinear programming, the constraints are not necessarily linear. Nonetheless, many of the same principles apply.

History

According to George Dantzig, the duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the lin…

Applications

In support vector machines (SVMs), the formulating the primal problem of SVMs as the dual problem can be used to implement Kernel trick, but the …

Here's what's really going on with the dual problem. (This is my attempt to answer my own question, over a year after originally asking it.) (A ve...Best answer · 150

I'll take a crack at a couple of these questions (some of them are hard and would require more thought). 1) Here's a nice economic interpretation o...52

Consider the problem $$ \begin{aligned} \mbox{min} \quad& f(x) \\ \mbox{subject to} \quad& x\le a\\ \end{aligned} $$ illustrated below and where $f...52

For #4 and #5, see "The concept of duality in convex analysis, and the characterization of the Legendre transform" by Artstein-Avidan and Milman.12

after reading and learning from this excellent discussion, I summarised an explanation based on my own background. If I made any mistakes, please c...9

A lot of great explanations. The easiest way to get in-depth knowledge, I think, however, is just to study chapter 5 of this book , written by Sta...5

There's a lot of great answers, but they seem to require some understanding of the problem already - so below I write a very quick and basic deduct...5

Here are some counter-examples to help you understand KKT conditions and strong duality. The answer is from my other post: https://math.stackexcha...4


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