# Invex optimization revisited

• Ksenia Bestuzheva
• Hassan Hijazi
Article

## Abstract

Given a non-convex optimization problem, we study conditions under which every Karush–Kuhn–Tucker (KKT) point is a global optimizer. This property is known as KT-invexity and allows to identify the subset of problems where an interior point method always converges to a global optimizer. In this work, we provide necessary conditions for KT-invexity in n dimensions and show that these conditions become sufficient in the two-dimensional case. As an application of our results, we study the Optimal Power Flow problem, showing that under mild assumptions on the variables’ bounds, our new necessary and sufficient conditions are met for problems with two degrees of freedom.

## Keywords

Convex optimization Invex optimization Boundary-invexity Optimal power flow

## Notations

$$\partial S$$

Boundary of a set S.

$$x_i$$

ith component of vector $$\mathbf {x}$$.

$$f_{x_{i}}^{\prime } = \frac{\partial f}{\partial x_i}$$

Partial derivative of f with respect to $$x_i$$.

$$||\mathbf {x}||$$

Euclidean norm of vector $$\mathbf {x}$$.

$$\mathbf {x} \cdot \mathbf {y}$$

The dot product of vectors $$\mathbf {x}$$ and $$\mathbf {y}$$.

$$\mathbf {x}^T$$

The transpose of vector $$\mathbf {x}$$.

$$\overline{AB}$$

A segment between points A and B.

$$2\mathbb {N}, ~2\mathbb {N}{+}1$$

The sets of even and odd numbers.

$$f'_-(x), f'_+(x)$$

Left and right derivatives of f.

sign(x)

The sign function.

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