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A multi-convex optimization problem is one in which the variables can be partitioned into sets over which the problem is convex when the other variables are fixed. Multi-convex problems are generally solved approximately using variations on alternating or cyclic minimization. Multi-convex problems arise in many applications, such as nonnegative matrix factorization, generalized low rank models, and structured control synthesis, to name just a few. In most applications to date the multi-convexity is simple to verify by hand. In this paper we study the automatic detection and verification of multi-convexity using the ideas of disciplined convex programming. We describe an implementation of our proposed method that detects and verifies multi-convexity, and then invokes one of the general solution methods.
In this paper we introduce disciplined convex-concave programming (DCCP), which combines the ideas of disciplined convex programming (DCP) with convex-concave programming (CCP). Convex-concave programming is an organized heuristic for solving nonconv
We present a composition rule involving quasiconvex functions that generalizes the classical composition rule for convex functions. This rule complements well-known rules for the curvature of quasiconvex functions under increasing functions and point
We introduce log-log convex programs, which are optimization problems with positive variables that become convex when the variables, objective functions, and constraint functions are replaced with their logs, which we refer to as a log-log transforma
We introduce Newton-ADMM, a method for fast conic optimization. The basic idea is to view the residuals of consecutive iterates generated by the alternating direction method of multipliers (ADMM) as a set of fixed point equations, and then use a nons
This paper considers the problem of minimizing a convex expectation function over a closed convex set, coupled with a set of inequality convex expectation constraints. We present a new stochastic approximation type algorithm, namely the stochastic ap