Convex optimization and dual problem

Is dual function always convex?
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)..
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.

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.

In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, 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 per…

Dual problem

Usually the term "dual problem" refers to the Lagrangian dual problem but other dual problems are used – for example, the Wolfe dual pr…

Linear case

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

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 pre…

Can convex optimization problems with generalized inequality constraints be solved easily?

• The optimality condition for diﬀerentiable f 0, given in §4

2 3, holds without any change

We will also see (in chapter 11) that convex optimization problems with generalized inequality constraints can often be solved as easily as ordinary convex optimization problems

The problem of finding the best lower bound: is called the dual problem associated with the Lagrangian defined above. It optimal value is the dual optimal value. As noted above, is concave. This means that the dual problem, which involves the maximization of with sign constraints on the variables, is a convex optimization problem.

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