Is Lagrangian dual convex?

Proof: From the definitions, we have g⋆ = supu≥0,v g(u, v) = supu≥0,v infx L(x, u, v). Although the primal problem is not required to be convex, the dual problem is always convex. The Lagrangian dual problem yields a lower bound for the primal problem. It always holds true that f⋆ ≥ g⋆, called as weak duality.

How do you know if your Lagrangian is concave?

A function f : C → R, where C is a convex set, is concave if −f is convex, or equivalently if f(αx + (1 − α)y) ≥ αf(x) + (1 − α)f(y), for each x, y ∈ C, and α ∈ [0, 1].

Why Lagrange dual function is concave?

When the Lagrangian is unbounded below in x, the dual function takes on the value −∞. 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).

How do you determine if a function is convex or concave?

For a twice-differentiable function f, if the second derivative, f ”(x), is positive (or, if the acceleration is positive), then the graph is convex (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward).

What is Lagrangian dual function?

Lagrangian duality theory refers to a way to find a bound or solve an optimization problem (the primal problem) by looking at a different optimization problem (the dual problem). A fundamental idea of the duality theory is that the dual of a dual linear program is the original primal linear program.

Is the Lagrangian the dual?

The Lagrangian dual problem is obtained by forming the Lagrangian of a minimization problem by using nonnegative Lagrange multipliers to add the constraints to the objective function, and then solving for the primal variable values that minimize the original objective function.

What is the Lagrangian dual function?

What does affine mean in math?

An affine function is a function composed of a linear function + a constant and its graph is a straight line. The general equation for an affine function in 1D is: y = Ax + c. An affine function demonstrates an affine transformation which is equivalent to a linear transformation followed by a translation.

Can a Lagrangian be negative?

The value of the Lagrange multiplier is the sensitivity of the constrained objective to (small) changes in the constraint δg. So if a Lagrange multiplier associated with an inequality constraint gj(x∗) ≤ 0 is computed as a negative value, it is subsequently set to zero.

Is Lagrangian convex or concave?

The Lagrangian as already pointed out in Arshak’s answer may or may not be convex, but the objective of the dual problem will be a concave function (or convex depending on whether the primal is a minimization or maximization).

What is the Lagrangian of a function?

In general, the Lagrangian is the sum of the original objective function and a term that involves the functional constraint and a ‘Lagrange multiplier’λ. Suppose we ignore the functional constraint and consider the problem of maximizing the Lagrangian, subject only to the regional constraint.

What is the Lagrange dual function of the primal problem?

The Lagrange dual function is g ( λ, ν) = inf x ∈ D L ( x, λ, ν) where D is the set of feasible points. This function is always concave regardless of the whether the primal problem was convex. The explanation for why this is so comes from properties of composition of convex functions. More info can be found in Stephen Boyd’s book.

What is the Lagrangian method of optimization?

The solution of a constrained optimization problem can often be found by using the so-calledLagrangian method. We define theLagrangianas