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3876 results in Optimization

Page 3 of 194

  • 5 - Online Gradient Descent

  • Tor Lattimore, Google DeepMind, London
  • Book:
    Bandit Convex Optimisation
    Published online:
    07 March 2026
    Print publication:
    19 March 2026, pp 50-60

  • Summary

    In this chapterwe introduce an idea that is ubiquitous in zeroth-order optimisation, which is to use a gradient-based algorithm but replace the true gradients with estimated gradients of a smoothed loss. Except for Section 5.4, we assume throughout this chapter that the constraint set contains a euclidean ball of unit radius, the losses are bounded, Lipschitz and there is no noise:

  • 1 - Introduction and Problem Statement

  • Tor Lattimore, Google DeepMind, London
  • Book:
    Bandit Convex Optimisation
    Published online:
    07 March 2026
    Print publication:
    19 March 2026, pp 1-13
    • Chapter
      • You have access
    • PDF
    • Export citation

  • Summary

    This book is about approximately solving problems of the form.

  • 3 - Mathematical Tools

  • Tor Lattimore, Google DeepMind, London
  • Book:
    Bandit Convex Optimisation
    Published online:
    07 March 2026
    Print publication:
    19 March 2026, pp 26-41

  • Summary

    The purpose of this chapter is to introduce the necessary tools from optimisation, convex geometry and convex analysis. You can safely skip this chapter, referring back as needed. The main concepts introduced are as follows:

  • 8 - Exponential Weights

  • Tor Lattimore, Google DeepMind, London
  • Book:
    Bandit Convex Optimisation
    Published online:
    07 March 2026
    Print publication:
    19 March 2026, pp 94-117

  • Summary

    We already saw an application of exponential weights to linear and quadratic bandits in Chapter 7. The same abstract algorithm can also be used for convex bandits but the situation is more complicated. Throughout this chapter we assume the losses are bounded and there is no noise:

  • 7 - Linear and Quadratic Bandits

  • Tor Lattimore, Google DeepMind, London
  • Book:
    Bandit Convex Optimisation
    Published online:
    07 March 2026
    Print publication:
    19 March 2026, pp 79-93

  • Summary

    Function classes like Fb are non-parametric. In this chapter we shift gears by studying two important parametric classes: Fb,lin and Fb,quad. The main purpose of this chapter is to use the machinery designed for linear bandits to prove an upper bound on the minimax regret for quadratic bandits. On the positive side the approach is both elementary and instructive. More negatively, the resulting algorithm is not computationally efficient. Before the algorithms and regret analysis we need three tools: covering numbers, optimal experimental design and the exponential weights algorithm.

  • 9 - Cutting Plane Methods

  • Tor Lattimore, Google DeepMind, London
  • Book:
    Bandit Convex Optimisation
    Published online:
    07 March 2026
    Print publication:
    19 March 2026, pp 118-144

  • Summary

    Like the bisection method (Chapter 4), cutting plane methods are most naturally suited to the stochastic setting. For the remainder of the chapter we assume the setting is stochastic and the loss function is bounded:

Games and Coalitions
  • Bridging Non-cooperative and Cooperative Approaches
  • Akira Okada
  • Published online:
    14 March 2026
    Print publication:
    30 April 2026
  • Coalition formation is an important problem in economics, politics, and a broad range of other social situations. Examples of coalitions range from those at the level of individuals (families, couples, teams, employers, workers) through to those at the level of organisations and countries (political parties, free trade agreements, environmental agreements, military alliances). Traditionally, game theory has been divided into non-cooperative and cooperative games. The former approach scrutinizes individuals' rational behaviour under a well-specified process of a game. The latter presents various cooperative solutions based on collective rationality. Games and Coalitions draws on both approaches, providing a bridge between cooperative and non-cooperative analyses of coalition formation. Offering a useful research monograph regarding the models, results and applications of non-cooperative coalitional bargaining theory, this book illustrates how game theory applies to various economic and political problems, including resource allocation, public goods, wage bargaining, legislative bargaining, and climate cooperation.

Bandit Convex Optimisation
  • Tor Lattimore
  • Published online:
    07 March 2026
    Print publication:
    19 March 2026
  • This comprehensive reference brings readers to the frontier of research on bandit convex optimization or zeroth-order convex optimization. The focus is on theoretical aspects, with short, self-contained chapters covering all the necessary tools from convex optimization and online learning, including gradient-based algorithms, interior point methods, cutting plane methods and information-theoretic machinery. The book features a large number of exercises, open problems and pointers to future research directions, making it ideal for students as well as researchers.

  • FLEXURAL GRAVITY WAVE INTERACTION BY MULTIPLE THIN POROELASTIC PLATES IN FINITE-DEPTH WATER

  • KOUSHIK NANDI, SELINA HOSSAIN, BIMAN SARKAR, SOUMEN DE
  • Journal:
    The ANZIAM Journal / Volume 68 / 2026
    Published online by Cambridge University Press:
    11 February 2026, e3

  • This study investigates the hydroelastic interaction of flexural gravity waves with multiple porous elastic plates of varying lengths in finite-depth water, employing an integral equation approach. The floating ice sheet is modelled as a flexible plate of uniform thickness, governed by the Euler–Bernoulli beam equation. The primary objective is to evaluate the effectiveness of porous elastic plates as wave barriers for shoreline protection in ice-covered regions. Within the framework of linearized theory, the problem is formulated as a boundary value problem (BVP) and solved using an eigenfunction expansion method with nonorthogonal eigenfunctions. The mode-coupling relation is utilized to transform the BVP into a system of Fredholm-type integral equations, which is subsequently solved using the multi-term Galerkin approximation technique with Chebyshev polynomials. The numerical analysis evaluates the reflection and transmission coefficients, hydrodynamic forces, and wave energy dissipation, with a particular focus on the influence of the permeability and flexibility of the submerged plates, along with other relevant parameters. Validation is conducted by comparing the results with those of previous studies under specific conditions. This research underscores the practical benefits of incorporating porosity and flexibility into the model, demonstrating improved wave reflection and energy dissipation. Additionally, the findings reveal that the thickness of the ice sheet plays a crucial role in optimizing breakwater performance. The research delivers key insights into mitigating wave-induced forces and offers a reliable framework for designing effective and sustainable coastal protection systems that safeguard shorelines from high waves.

Page 3 of 194