Convex optimization

About: Convex optimization is a research topic. Over the lifetime, 24906 publications have been published within this topic receiving 908795 citations. The topic is also known as: convex optimisation.

FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation

Abstract: FORTRAN 77 codes SONEST and CONEST are presented for estimating the 1-norm ( or the infinity-norm) of a real or complex matrix, respectively. The codes are of wide applicability in condition estimation since explicit access to the matrix, A, is not required; instead, matrix-vector products Ax and ATx are computed by the calling program via a reverse communication interface. The algorithms are based on a convex optimization method for estimating the 1-norm of a real matrix devised by Hager. We derive new results concerning the behavior of Hager's method, extend it to complex matrices, and make several algorithmic modifications in order to improve the reliability and efficiency.

Trading Regret for Efficiency: Online Convex Optimization with Long Term Constraints

Abstract: In this paper we propose a framework for solving constrained online convex optimization problem. Our motivation stems from the observation that most algorithms proposed for online convex optimization require a projection onto the convex set $\mathcal{K}$ from which the decisions are made. While for simple shapes (e.g. Euclidean ball) the projection is straightforward, for arbitrary complex sets this is the main computational challenge and may be inefficient in practice. In this paper, we consider an alternative online convex optimization problem. Instead of requiring decisions belong to $\mathcal{K}$ for all rounds, we only require that the constraints which define the set $\mathcal{K}$ be satisfied in the long run. We show that our framework can be utilized to solve a relaxed version of online learning with side constraints addressed in \cite{DBLP:conf/colt/MannorT06} and \cite{DBLP:conf/aaai/KvetonYTM08}. By turning the problem into an online convex-concave optimization problem, we propose an efficient algorithm which achieves $\tilde{\mathcal{O}}(\sqrt{T})$ regret bound and $\tilde{\mathcal{O}}(T^{3/4})$ bound for the violation of constraints. Then we modify the algorithm in order to guarantee that the constraints are satisfied in the long run. This gain is achieved at the price of getting $\tilde{\mathcal{O}}(T^{3/4})$ regret bound. Our second algorithm is based on the Mirror Prox method \citep{nemirovski-2005-prox} to solve variational inequalities which achieves $\tilde{\mathcal{\mathcal{O}}}(T^{2/3})$ bound for both regret and the violation of constraints when the domain $\K$ can be described by a finite number of linear constraints. Finally, we extend the result to the setting where we only have partial access to the convex set $\mathcal{K}$ and propose a multipoint bandit feedback algorithm with the same bounds in expectation as our first algorithm.

Convex Optimization of Power Systems

Abstract: Optimization is ubiquitous in power system engineering. Drawing on powerful, modern tools from convex optimization, this rigorous exposition introduces essential techniques for formulating linear, second-order cone, and semidefinite programming approximations to the canonical optimal power flow problem, which lies at the heart of many different power system optimizations. Convex models in each optimization class are then developed in parallel for a variety of practical applications like unit commitment, generation and transmission planning, and nodal pricing. Presenting classical approximations and modern convex relaxations side-by-side, and a selection of problems and worked examples, this is an invaluable resource for students and researchers from industry and academia in power systems, optimization, and control.

Variable metric bundle methods: from conceptual to implementable forms

Abstract: To minimize a convex function, we combine Moreau-Yosida regularizations, quasi-Newton matrices and bundling mechanisms. First we develop conceptual forms using "reversal" quasi-Newton formulae and we state their global and local convergence. Then, to produce implementable versions, we incorporate a bundle strategy together with a "curve-search". No convergence results are given for the implementable versions; however some numerical illustrations show their good behaviour even for large-scale problems.

Network Equilibrium of Coupled Transportation and Power Distribution Systems

Abstract: This paper presents a holistic modeling framework for the interdependent transportation network and power distribution network. From a system-level perspective, on-road fast charging stations would simultaneously impact vehicle routing in the transportation system and load flows in the distribution system, therefore tightly couple the two systems. In this paper, a dedicated traffic user equilibrium model is proposed to describe the steady-state distribution of traffic flows comprised of gasoline vehicles and electric vehicles. It encapsulates route selections, charging opportunities, electricity prices, and individual rationalities of minimum travel expense in a convex traffic assignment problem over an extended transportation network. An adaptive path generation oracle is suggested to solve the problem in a tractable manner. Economic operation of the power distribution system is formulated as an alternating current optimal power flow problem. Convex relaxation is performed. The optimal generation dispatch and nodal electricity prices can be computed from a second-order cone program. It is revealed that an equilibrium state will emerge due to the rational behaviors in the coupled systems, which is characterized via a fixed-point problem. A best-response decomposition algorithm is suggested to identify the network equilibrium through iteratively solving the traffic assignment problem and the optimal power flow problem, both of which entail convex optimization. Illustrative examples are presented to validate related concepts and methods.