[Paper Review] Homogeneous Cone Complementarity Problems and $P$ Properties

Lingchen Kong, Levent Tunçel+1 more|arXiv (Cornell University)|Apr 11, 2009

Advanced Topics in Algebra29 references3 citations

TL;DR

This paper establishes existence and uniqueness conditions for solutions to homogeneous cone complementarity problems (HCCP) using generalized P-properties in the context of T-algebraic structures of homogeneous cones. It proves that if a continuous function satisfies either the order-P₀ and R₀ or P₀ and R₀ properties, then HCCP has a solution for every q; further, monotone functions with the trace-P or uniform-trace-P property ensure global uniqueness and error bounds.

ABSTRACT

We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem (HCCP). Employing the $T$-algebraic characterization of homogeneous cones, we generalize the $P, P_0, R_0$ properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of HCCP. We prove that if a continuous function has either the order-$P_0$ and $R_0$, or the $P_0$ and $R_0$ properties then all the associated HCCPs have solutions. In particular, if a continuous function has the trace-$P$ property then the associated HCCP has a unique solution (if any); if it has the uniform-trace-$P$ property then the associated HCCP has the global uniqueness (of the solution) property (GUS). We present a necessary condition for a nonlinear transformation to have the GUS property. Moreover, we establish a global error bound for the HCCP with the uniform-trace-$P$ property. Finally, we study the HCCP with the relaxation transformation on a $T$-algebra and automorphism invariant properties for homogeneous cone linear complementarity problem.

Motivation & Objective

• To extend P-properties from standard nonlinear complementarity problems to the broader setting of homogeneous cone complementarity problems (HCCP).
• To establish sufficient conditions for the existence of solutions to HCCP using generalized P₀ and R₀ properties.
• To characterize global uniqueness of solutions (GUS property) for HCCP under monotonicity and uniform-trace-P conditions.
• To derive a global error bound for HCCP under the uniform-trace-P property, generalizing known results from NCP and SCCP.
• To extend theoretical results to arbitrary convex cones, suggesting future research directions in broader conic optimization.

Proposed method

• Employing the T-algebraic characterization of homogeneous cones to analyze metric projections and their properties in finite-dimensional inner product spaces.
• Introducing new P-properties—order-P₀, trace-P, uniform-trace-P, and their variants—for continuous functions F: H → H in the HCCP setting.
• Using degree theory to prove existence of solutions when F satisfies P₀ and R₀ or order-P₀ and R₀ properties.
• Applying metric projection identities and spectral decomposition via Jordan frames to derive error bound estimates.
• Leveraging Lipschitz continuity of F and norm inequalities in T-algebras to bound the distance between iterates and solutions.
• Deriving a global error bound by combining the uniform-trace-P property with projection stability and norm contraction arguments.

Experimental results

Research questions

• RQ1 Under what conditions on a continuous function F does the homogeneous cone complementarity problem (HCCP) have a solution for every q ∈ H?
• RQ2 How can the classical P-property and P₀-property be generalized to the setting of homogeneous cones?
• RQ3 What conditions ensure the global uniqueness of solution (GUS) for HCCP when F is monotone?
• RQ4 Can a global error bound be established for HCCP under the uniform-trace-P property?
• RQ5 To what extent do these results generalize to arbitrary convex cones beyond homogeneous cones?

Key findings

• If a continuous function F has both the order-P₀ and R₀ properties, then the HCCP(F, q) has a solution for every q ∈ H.
• If F has both the P₀ and R₀ properties, then HCCP(F, q) is solvable for all q ∈ H, extending classical NCP results to HCCP.
• If F is monotone and has the trace-P property, then HCCP(F, q) has a unique solution (if any), generalizing uniqueness results from symmetric cones.
• If F is monotone and has the uniform-trace-P property, then HCCP(F, q) satisfies the globally uniquely solvability (GUS) property.
• A global error bound is established for HCCP under the uniform-trace-P property: ‖x − x*‖ ≤ C ⋅ max_i |⟨(F(x) − F(x*))(x − x*), e_i⟩|, with C depending on Lipschitz constant and cone structure.
• The error bound generalizes known results for NCP and SCCP, and the proof relies on spectral decomposition and projection stability in T-algebras.