Monotonic Optimization

Globally Optimal Energy-Efficient Power Control and Receiver Design in Wireless Networks

“…Nevertheless, we remark that the complexity of monotonic optimization methods is still exponential in the number of variables. However, as already observed, it is much lower than general global optimization methods, which do not exploit any monotonicity structure [21]. This makes monotonic optimization attractive for the development of an off-line framework to benchmark any suboptimal method for solving (24).…”

“…This optimization framework can solve optimization problems where the utility and constraints are monotonically increasing/decreasing functions of the variables, even if no convexity assumption is fulfilled. The polyblock algorithm [21] and the branch-reduce-and-bound (BRB) algorithm [22] are two key algorithms for globally solving these problems. These algorithms have recently been used to solve different non-convex problems in communication and networking systems; for example, joint power control and scheduling [23] and sum rate maximization with multi-antenna transmitters [24].…”

“…Section III-A gives a short review of fractional programming theory [33] and also provides some background (see [34] for more details) to understand the complexity arguments mentioned at the end of Section II. Then, Section III-B gives an overview of monotonic optimization [21], [22]. Finally, Section III-C briefly discusses the framework of sequential optimization.…”

“…Monotonic optimization is a relatively recent global optimization framework, which exploits monotonicity or hidden monotonicity structures in the objective and constraints to reduce computational complexity and provide a guaranteed convergence [21], [22]. The basic idea is that if the objective to be maximized is increasing in all optimization variables, then it is not necessary to explore the complete feasible set of the problem, but only its outer boundary.…”

“…1 Such a class of problems can be solved with polynomial-time complexity [37], [38] Definition A given function h : R N → R defines a normal or a conormal set if the following results hold true: The main result of monotonic optimization theory states that the solution to (24) lies on the upper boundary of S ∩ S c [21,Proposition 7]. Therefore, methods like the polyblock algorithm [21] and the BRB algorithm [22] can be used to globally solve (24) by searching only on the upper boundary of the feasible set, thus drastically simplifying the problem. Nevertheless, we remark that the complexity of monotonic optimization methods is still exponential in the number of variables.…”