Convex envelopes generated from finitely many compact convex sets

ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations

“…The polymorphic strategy illustrated in Figure 2(a) allows us to easily integrate many disparate term types into a single framework; we also avoid duplicating code for similar types. Observe that specialized terms which are not at this writing integrated into the framework (e.g., [61,71,72,86,129,139]) could be easily added to Figure 2(a); we anticipate further advances in the area of term-specific underestimators and have architected ANTIGONE accordingly.…”

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

“…The complementary horizontal, term-based data structures easily admit multivariable relaxations that are specifically designed for particular functional forms. For example, beyond the convex, bilinear, trilinear, fractional, fractional trilinear, univariate concave, and general nonconvex terms as introduced by Adjiman et al [33,34], underestimators have been introduced or improved for: fractional terms [31,36,54]; trilinear terms [55,56]; quadrilinear terms [57]; odd degree monomials [58]; signomial terms [8,48,50]; low-dimensional edge-concave terms [59][60][61][62]; submodular functions [63]; and interesting products [36,[64][65][66][67].…”

“…The MISO framework transforms a factorable programming tree into a flattened expression tree to capitalize on the development of tight convex underestimators for specific classes of nonlinear terms (e.g., [8,31,36,48,50,[54][55][56][57][58][59][60][61][62][63][64][65][66][67]). This hybrid approach reformulates towards multivariable terms with specialized underestimators while maintaining a tree-like representation of powers that cannot be distributed and convex operators that can be exploited by dynamic cut generation.…”

Extended formulations for convex envelopes

“…In general, problem (1) is a nonconvex problem which is extremely difficult to solve. Over the last decade, many articles dealing with the computation of envelopes for several classes of functions have been published (e.g., see [5,[14][15][16]20,21,35,37]). Usually, analytical and geometric properties of the functions are used to reduce the complexity and to derive alternative formulations of problem (1) that are much more tractable.…”

“…Convex envelopes are only known for subclasses and have been recently deduced by Khajavirad and Sahinidis [15,16]. They further assume that f (x, y) is decomposable into the form g(x) • h(y), and that g is submodular and convex-extendable from the vertices of [l x , u x ].…”