# Convex Extensions and Relaxation Strategies

• Mohit Tawarmalani
• Nikolaos V. Sahinidis
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 65)

## Synopsis

Central to the efficiency of global optimization methods for nonconvex mathematical programs is the capability to construct tight convex relaxations. In this chapter, we develop the theory of convex extensions of lower semicontinuous (l.s.c.) functions and illustrate its use in building convex envelopes of nonconvex mathematical programs. The techniques developed here amount to a recipe that can be used to construct closed-form expressions of convex and concave envelopes of many classes of functions.

We define a convex extension of a lower semicontinuous nllletion to be a eonvex function that is identical to the given function over a prespecified subset of its domain. Convex extensions are not necessarily eonstruetible or unique. We identify conditions under whieh a eonvex extension can be construeted. When multiple eonvex extensions exist, we characterize the tightest eonvex extension in a well-defined sense. Using the notion of a generating set, we establish conditions under whieh the tightest eonvex extension is the convex envelope. Then, we employ convex extensions to develop a construetive technique for deriving convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of x/y over a reet angle and prove that the extensions theory provides eonvex relaxations that are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms.

## Keywords

Concave Function Convex Relaxation Lower Semicontinuous Function Convex Envelope Bilinear Term
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