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Positive Definite Tensors to Nonlinear Complementarity Problems
Maolin Che, Liqun Qi, Yimin Wei
TL;DR
The paper studies existence and uniqueness for nonlinear complementarity problems generated by structured tensor mappings. It formulates tensor-polynomial NCPs, applies Jacobian and optimization-based conditions, and proves unique solutions under diagonalizability and positive definiteness, with stated scope restrictions.
Problem
The paper asks whether tensor-based nonlinear complementarity problems possess existence and uniqueness properties analogous to those known for linear complementarity problems.
Method
The paper represents NCP mappings as multivariate tensor polynomials and uses Jacobian bounds, nonlinear-programming conditions, and tensor structure to analyze solvability.
Results
Diagonalizable positive definite tensors yield a unique solution for NCP(q,A), and corresponding diagonalizable positive definite tensor families yield a unique GNCP solution.
Takeaways & Limitations
The results establish solvability conclusions for structured tensor complementarity problems, including unique-solution results under diagonalizability and positive definiteness.
Abstract
from arXiv · showhide
The main purpose of this note is to investigate some kinds of nonlinear complementarity problems (NCP). For the structured tensors, such as, symmetric positive definite tensors and copositive tensors, we derive the existence theorems on a solution of these kinds of nonlinear complementarity problems. We prove that a unique solution of the NCP exists under the condition of diagonalizable tensors.
1 Introduction
The introduction formulates nonlinear complementarity problems and places tensor-based polynomial mappings within the broader complementarity and variational-inequality framework. It motivates questions about existence and uniqueness for structured tensors, including P tensors and diagonalizable tensors.
- NCP(F) seeks x∗∈R^n satisfying nonnegativity and complementarity conditions for F.
- When F(x)=q+Mx is affine, NCP(F) reduces to the linear complementarity problem LCP(q,M).
- Variational inequality problems generalize NCPs, with K={x:x≥0} yielding an equivalent formulation.
- The introduction asks whether P-tensor analogues retain the unique-solution property known for P matrices.
- The note studies NCPs whose component mappings are multivariate polynomials represented through tensor-vector products.
2 Preliminaries
The preliminaries define tensor operations, symmetry, positivity, copositivity, diagonalizability, and regularity, then state foundational existence and uniqueness results used for the paper’s complementarity questions.
- Tensor classes: Symmetric even-order tensors are classified as positive definite, copositive, or strictly copositive through the polynomial Ax^m.
- Mapping conditions: A d-regular mapping is characterized by the absence of nonzero solutions to an augmented nonlinear complementarity system.
- Tensor classes: A diagonalizable tensor has the form A=D×_1B×_2B···×_mB, with det(B)≠0 and diagonal D.
- Existence and uniqueness: If F is continuously differentiable and all principal minors of its Jacobian remain between δ and δ^-1, NCP(F) has a unique solution.
- Existence and uniqueness: A continuous mapping that is positively homogeneous and d-regular has a nonempty, compact NCP solution set.
- Problem description: The paper poses tensor NCP and generalized NCP questions, treating the former as a special case of the latter.
3 Main results
The paper derives existence and uniqueness results for tensor nonlinear complementarity problems by linking them to nonlinear programming conditions and Jacobian properties. For structured tensors, diagonalizability yields uniqueness, while positive definiteness and strict copositivity yield nonempty compact solution sets for both considered problem classes.
- Necessary conditions: A local solution of the nonlinear program yields multipliers satisfying conditions that connect feasibility, complementarity, and a solution of NCP(q, A).The argument uses first-order Karush-Kuhn-Tucker conditions and inner-product identities.
- Existence for Question 1: If a nonzero local solution satisfies positive definiteness of Ax^(m−2), it solves NCP(q, A).This provides the existence bridge from the nonlinear programming formulation to the complementarity problem.
- Solving Question 1: Positive definiteness makes F(x)=Ax^(m−1)+q strongly copositive, whereas copositivity of A transfers to corresponding copositivity of F.For positive definite diagonalizable tensors, the Jacobian ∇F(x) is positive definite for every nonzero x.
- Solving Question 1: For A ∈ ST_m,n, diagonalizability and positive definiteness give a unique solution, while positive definiteness alone gives a nonempty, compact solution set.Strict copositivity with respect to R^n_+ also gives a nonempty, compact solution set.
- Solving Question 2: For Question 2, diagonalizable positive definite tensors yield a unique GNCP(q,{A_k}) solution, while positive definiteness or strict copositivity yields a nonempty, compact solution set.The stated assumptions include principal-minor bounds for A^(m/2) in the uniqueness result and strict copositivity conditions for existence.
- Solving Question 2: The Question 2 assumptions can be weakened when A^(m/2) is square, although the paper does not consider all such reductions.Theorem 7 specializes this setting to k ∈ [m/2−1].
4 Conclusion
The note studies existence and uniqueness for nonlinear complementarity problems involving structured tensors, including positive definite and diagonalizable cases. It also identifies an open condition question concerning bounded principal minors of Ax^{m−2}.
- The main task is to establish existence and uniqueness of solutions for NCP(q, A) with structured tensors.
- Positive definiteness alone is conjectured to guarantee a unique solution for NCP(q, A) in the stated theorem case.
- If FEA(q, A) is nonempty, the associated nonlinear program is conjectured to have an optimal solution x* and suitable multipliers.
- For m = 2, the existence statement reduces to the known quadratic-program result for linear complementarity problems, but counterexamples invalidate the analogous general nonlinear-programming conjecture.
- An open question asks which conditions on A ensure that all principal minors of Ax^{m−2} remain between δ and δ−1 for every nonzero x.