First-Order Methods for Convex Optimization

Pavel Dvurechensky, Shimrit Shtern, Mathias Staudigl

Research output: Contribution to journalArticlepeer-review


First-order methods for solving convex optimization problems have been at the forefront of mathematical optimization in the last 20 years. The rapid development of this important class of algorithms is motivated by the success stories reported in various applications, including most importantly machine learning, signal processing, imaging and control theory. First-order methods have the potential to provide low accuracy solutions at low computational complexity which makes them an attractive set of tools in large-scale optimization problems. In this survey, we cover a number of key developments in gradient-based optimization methods. This includes non-Euclidean extensions of the classical proximal gradient method, and its accelerated versions. Additionally we survey recent developments within the class of projection-free methods, and proximal versions of primal-dual schemes. We give complete proofs for various key results, and highlight the unifying aspects of several optimization algorithms.

Original languageEnglish
Article number100015
JournalEURO Journal on Computational Optimization
StatePublished - Jan 2021


  • Bregman Divergence
  • Composite Optimization
  • Convergence Rate
  • Convex Optimization
  • First-Order Methods
  • Numerical Algorithms
  • Proximal Mapping
  • Proximity Operator

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Management Science and Operations Research
  • Control and Optimization
  • Computational Mathematics


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