Duality for Nonlinear Programming

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Infimum

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We define the dual objective function $q:{\mathbb{R}}^n\to \mathbb{R}$ as follows

$$q(\lambda )\triangleq \underset{x}{\inf }\mathcal{L}(x,\lambda )$$

Where $\mathcal{L}=f(x)-{\lambda }^{T}c(x)$ is the Lagrangian. We also define the domain of $q$ as the set of $\lambda$ values for which $q$ is finite, that is

$$\mathcal{D}\triangleq \{\lambda |q(\lambda )>-\infty \}$$

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The dual problem can be used to obtain a lower bound on the optimal value of the objective for the primal problem.

Restrictions

No equality constraints The objective $f$ and the negatives of the inequality constrains, $c_i$, makes our problem a Convex Optimization Problem

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