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Quartic Regularity.

Yurii Nesterov1,2

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Summary
This summary is machine-generated.

New second-order optimization methods achieve linear convergence for convex quartic polynomials. This framework extends to general convex problems with quartic regularity, offering improved solution efficiency.

Keywords:
Composite convex minimizationGlobal complexity boundsHigh-order proximal-point methodsSecond-order methods

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Area of Science:

  • Optimization Theory
  • Numerical Analysis
  • Convex Optimization

Background:

  • Minimizing convex quartic polynomials is a fundamental problem in optimization.
  • Existing methods may lack efficiency for problems with specific regularity conditions.
  • High-order proximal-point schemes offer advanced convergence properties.

Purpose of the Study:

  • To develop novel, linearly convergent second-order methods for convex quartic polynomial minimization.
  • To design optimization schemes for general convex problems exhibiting quartic regularity.
  • To explore the application of these methods within high-order proximal-point frameworks.

Main Methods:

  • Development of second-order optimization algorithms based on quartic regularization.
  • Application of a modified Damped Newton Method for problems with quartic regularity.
  • Integration of these methods into high-order proximal-point schemes.

Main Results:

  • Achieved global linear convergence rate for quartic regularization of Damped Newton Method.
  • Demonstrated applicability to general convex problems satisfying quartic regularity.
  • Derived methods with convergence rates of Õ(k⁻ᵖ) for p=3, 4, or 5.

Conclusions:

  • The proposed second-order methods provide efficient solutions for convex quartic polynomials.
  • The quartic regularity condition enables the design of effective optimization schemes.
  • These advancements contribute to the theory and practice of high-order optimization.