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This study introduces a new, exact reformulation for nonconvex quadratic optimization problems with complementarity constraints using mild conditions. This method enables exact sparse solutions in optimization, linking quadratic problems to copositive optimization.

Keywords:
Complementarity constraintsConic relaxationCopositive optimizationQuadratic optimizationSparsity

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Area of Science:

  • Optimization Theory
  • Mathematical Programming
  • Operations Research

Background:

  • Nonconvex quadratic optimization problems with complementarity constraints are challenging.
  • Existing methods often require branching or large constants, limiting practical application.
  • Interpretability and sparsity are crucial in many optimization applications.

Purpose of the Study:

  • To establish an exact completely positive reformulation for a class of quadratic optimization problems with complementarity constraints.
  • To derive conditions for strong conic duality for the reformulated problem.
  • To demonstrate the application of this approach to achieving interpretable sparse solutions.

Main Methods:

  • Developing purely continuous models that avoid branching and large constants.
  • Establishing mild conditions based solely on constraints for the reformulation.
  • Investigating the strong conic duality between the completely positive problem and its dual.

Main Results:

  • An exact completely positive reformulation is established under new, mild conditions.
  • Conditions for strong conic duality are identified.
  • The approach successfully links quadratic problems with exact sparsity terms to copositive optimization.

Conclusions:

  • The proposed method offers an efficient and exact approach for solving specific quadratic optimization problems.
  • The framework is applicable to problems like sparse least-squares regression under linear constraints.
  • Numerical comparisons indicate favorable objective function values compared to other approximation methods.