nonconvexnonconvex

Nonconvexnonconvex is not a standard term in mathematics or optimization literature. It may appear as a repetition of the word nonconvex or as a coined label in informal discussions, research notes, or project names. In most uses, it would be interpreted as referring to a problem that exhibits nonconvexity in multiple aspects.

It could denote nonconvexity in more than one component of a model, for example nonconvex objective functions together with nonconvex constraint sets, or a combination of nonconvex domains and nonconvex penalties. In some cases it is used jokingly to stress the difficulty of a problem by emphasizing nonconvexity twice.

Nonconvex problems, in general, do not guarantee a global optimum found by local search. They can have many local minima, saddle points, and flat regions. The landscape depends on the objective and constraints, as well as on regularization and initialization. Verification of global optimality is typically hard.

Examples of nonconvex problems commonly encountered include training deep neural networks, certain matrix factorization tasks, phase retrieval, and clustering with nonconvex loss functions. Approaches to handle nonconvexity include convex relaxations, global optimization techniques, heuristic methods, good initialization, and problem-specific structural assumptions that enable tractable analysis or guarantees in special cases.

See also: nonconvex optimization, convex relaxation, global optimization, landscape analysis.