The exactness of the penalty function for a class of mathematical programs with generalized complementarity...

Area of Science:

• Optimization Theory
• Mathematical Programming
• Computational Mathematics

Background:

• Mathematical programs with generalized complementarity constraints (MPGCC) are challenging due to disjunctive feasible regions.
• The penalty method is commonly used for computation, but its exactness for MPGCC remains unclear.
• Existing tools struggle to analyze the exactness of penalty functions for certain MPGCC instances.

Purpose of the Study:

• To analyze the exactness of the penalty method for a class of multi-affine MPGCCs.
• To establish theoretical guarantees for the penalty method in MPGCC.
• To extend existing exactness results for MPCC to the multi-block MPGCC context.

Main Methods:

• Investigating a specific class of MPGCCs with multi-affine objective functions.
• Developing theoretical tools to prove the exactness of the penalty function.
• Analyzing the relationship between the original MPGCC and its penalty subproblem.

Main Results:

• An instance of MPGCC is presented where existing tools fail to prove penalty function exactness.
• Exactness results for the penalty method are established under mild conditions for the studied MPGCC class.
• The findings generalize and encompass existing exactness results for traditional MPCC.

Conclusions:

• The penalty method can be exact for a significant class of MPGCCs, including those with multi-affine objectives.
• The established results provide a theoretical foundation for using penalty methods in solving complex complementarity problems.
• This work advances the computational approaches for problems in areas like optimal transport and network pricing.