Biconvex optimization

Biconvex optimization is a generalization of convex optimization where the objective function and the constraint set can be biconvex. There are methods that can find the global optimum of these problems.^[1]^[2]

A set $B\subset X\times Y$ is called a biconvex set on $X\times Y$ if for every fixed $y\in Y$ , $B_{y}=\{x\in X:(x,y)\in B\}$ is a convex set in $X$ and for every fixed $x\in X$ , $B_{x}=\{y\in Y:(x,y)\in B\}$ is a convex set in $Y$ .

A function $f(x,y):B\to {\mathbb {R}}$ is called a biconvex function if fixing $x$ , $f_{x}(y)=f(x,y)$ is convex over $Y$ and fixing $y$ , $f_{y}(x)=f(x,y)$ is convex over $X$ .

A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating $x,y$ by fixing one of them and solving the corresponding convex optimization problem.^[1]

References

1 2 Gorski, Jochen; Pfeuffer, Frank; Klamroth, Kathrin (22 June 2007). "Biconvex sets and optimization with biconvex functions: a survey and extensions" (PDF). Mathematical Methods of Operations Research. 66 (3): 373–407. doi:10.1007/s00186-007-0161-1.
↑ Floudas, Christodoulos A. (2000). Deterministic global optimization : theory, methods, and applications. Dordrecht [u.a.]: Kluwer Academic Publ. ISBN 978-0-7923-6014-8.

Optimization: Algorithms, methods, and heuristics

Unconstrained nonlinear: Methods calling …

… functions

… and gradients

Convergence	Trust region Wolfe conditions
Quasi–Newton	BFGS and L-BFGS DFP Symmetric rank-one (SR1)
Other methods	Gauss–Newton Gradient Levenberg–Marquardt Conjugate gradient Truncated Newton

… and Hessians

Newton's method

Graph of a strictly concave quadratic function with unique maximum.

Constrained nonlinear

General	Barrier methods Penalty methods
Differentiable	Augmented Lagrangian methods Sequential quadratic programming Successive linear programming

Convex optimization

Convex
minimization

Linear and
quadratic

Interior point	Affine scaling Ellipsoid algorithm of Khachiyan Projective algorithm of Karmarkar
Basis-exchange	Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm of Lemke

Combinatorial

Paradigms

Graph
algorithms

Minimum spanning tree	Bellman–Ford Borůvka Dijkstra Floyd–Warshall Johnson Kruskal

Network flows

Metaheuristics
Evolutionary algorithm Hill climbing Local search Simulated annealing Tabu search

Software

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