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Global optimization in Hilbert space.

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Summary
This summary is machine-generated.

A new complete-search algorithm finds global optima for complex non-convex optimization problems. This method guarantees convergence to a near-optimal solution in finite time, even for infinite-dimensional problems.

Keywords:
Branch-and-liftComplete searchComplexity analysisConvergence analysisInfinite-dimensional optimization

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Area of Science:

  • Optimization Theory
  • Computational Mathematics

Background:

  • Non-convex optimization problems are prevalent in science and engineering.
  • Finding global optima for these problems is often computationally intractable.
  • Existing methods may converge to local optima or require infinite time.

Purpose of the Study:

  • To develop a complete-search algorithm for global optimality in non-convex optimization.
  • To analyze the algorithm's performance and convergence properties.
  • To demonstrate the algorithm's applicability to infinite-dimensional problems.

Main Methods:

  • A novel complete-search algorithm is proposed.
  • Worst-case run-time bounds are determined under regularity conditions.
  • Convergence analysis is performed for bounded Hilbert spaces.
  • The algorithm is illustrated using a calculus of variations problem.

Main Results:

  • The algorithm achieves global optimality for a class of non-convex problems.
  • Run-time bounds are independent of the number of variables, including infinite dimensions.
  • Guaranteed convergence to an epsilon-suboptimal global solution in finite time.
  • Successful application to a calculus of variations problem.

Conclusions:

  • The proposed algorithm offers a robust method for solving challenging non-convex optimization problems to global optimality.
  • Finite-time convergence to a specified tolerance is proven, even for infinite-dimensional cases.
  • This work advances the field of global optimization with practical implications.