Feasibility and solvability of Lyapunov-type linear programming over symmetric cones

In this paper, we consider the Lyapunov-type linear programming and its dual over symmetric cones. By introducing and characterizing the generalized inverse of Lyapunov operator in Euclidean Jordan algebras, we establish two kinds of Lyapunov-type Farkas’ lemmas to exhibit feasibilities of the corresponding primal and dual programming problems, respectively. As one of the main results, we show that the feasibilities of the primal and dual problems lead to the solvability of the primal problem and zero duality gap under some mild condition. In this case, we obtain that any solution to the pair of primal and dual problems is equivalent to the solution of the corresponding KKT system.     

Smoothing algorithms for complementarity problems over symmetric cones

There recently has been much interest in studying optimization problems over symmetric cones. In this paper, we first investigate a smoothing function in the context of symmetric cones and show that it is coercive under suitable assumptions. We then extend two generic frameworks of smoothing algorithms to solve the complementarity problems over symmetric cones, and prove the proposed algorithms are globally convergent under suitable assumptions. We also give a specific smoothing Newton algorithm which is globally and locally quadratically convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for second-order cone complementarity problems are reported.     

On the Lipschitzian property in linear complementarity problems over symmetric cones

Let V be a Euclidean Jordan algebra with symmetric cone K. We show that if a linear transformation L on V has the Lipschitzian property and the linear complementarity problem LCP(L,q) over K has a solution for every invertible q∈V, then 〈L(c),c〉>0 for all primitive idempotents c in V. We show that the converse holds for Lyapunov-like transformations, Stein transformations and quadratic representations. We also show that the Lipschitzian Q-property of the relaxation transformation R_A on V implies that A is a P-matrix.     

On the coerciveness of some merit functions for complementarity problems over symmetric cones

One of the popular solution methods for the complementarity problem over symmetric cones is to reformulate it as the global minimization of a certain merit function. An important question to be answered for this class of methods is under what conditions the level sets of the merit function are bounded (the coerciveness of the merit function). In this paper, we introduce the generalized weak-coerciveness of a continuous transformation. Under this condition, we prove the coerciveness of some merit functions, such as the natural residual function, the normal map, and the Fukushima-Yamashita function for complementarity problems over symmetric cones. We note that this is a much milder condition than strong monotonicity, used in the current literature.     

Construction of proximal distances over symmetric cones

Abstract

This paper is devoted to the study of proximal distances defined over symmetric cones, which include the non-negative orthant, the second-order cone and the cone of positive semi-definite symmetric matrices. Specifically, our first aim is to provide two ways to build them. For this, we consider two classes of real-valued functions satisfying some assumptions. Then, we show that its corresponding spectrally defined function defines a proximal distance. In addition, we present several examples and some properties of this distance. Taking into account these properties, we analyse the convergence of proximal-type algorithms for solving convex symmetric cone programming (SCP) problems, and we study the asymptotic behaviour of primal central paths associated with a proximal distance. Finally, for linear SCP problems, we provide a relationship between the proximal sequence and the primal central path.     

Nondifferentiable multiobjective symmetric dual programs over cones

Wolfe and Mond-Weir type nondifferentiable multiobjective symmetric dual programs are formulated over arbitrary cones and appropriate duality theorems are established under K-preinvexity/K-convexity/pseudoinvexity assumptions.     

Complementarity properties of the Lyapunov transformation over symmetric cones

The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX +XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V . Given any element a ∈ V , we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0 -property for La and show that La has the R0 -property if and only if a is invertible. Finally, we provide La with some characterizations of the E0 -property and the nondegeneracy property.     

A new class of smoothing complementarity functions over symmetric cones

A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Fréchet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones.     

A new infeasible-interior-point algorithm for linear programming over symmetric cones

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ₁, τ₂, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r^1.5logε^?1) for the Nesterov-Todd (NT) direction, and O(r²logε^?1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε > 0 is the required precision. If starting with a feasible point (x⁰, y⁰, s⁰) in N(τ₁, τ₂, η), the complexity bound is \(O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)\) for the NT direction, and O(rlogε^?1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.     

Lyapunov-type inequalities for one-dimensional Minkowski-curvature problems

Lyapunov-type inequalities are estimated to give necessary conditions for the existence of positive solutions for several types of scalar equations and systems of one-dimensional Minkowski-curvature problems with singular weights.     

不可微多目标规划的高阶对称对偶性

本文研究锥约束不可微多目标规划的Mond-Weir 型高阶对称对偶问题. 本文指出Agarwal 等人(2010) 和Gupta 等人(2010) 工作的不足, 给出规划问题的强对偶和逆对偶定理.     

A Mizuno-Todd-Ye predictor-corrector infeasible-interior-point method for linear programming over symmetric cones

We propose a new general type of splitting methods for accretive operators in Banach spaces. We then give the sufficient conditions to guarantee the strong convergence. In the last section, we apply our results to the minimization optimization problem and the linear inverse problem including the numerical examples.     

Off-diagonal estimates of some Bergman-type operators of tube domains over symmetric cones

We obtain some necessary and sufficient conditions for the boundedness of a family of positive operators defined on symmetric cones, we then deduce off-diagonal boundedness of associated Bergman-type operators in tube domains over symmetric cones.     

Extension of primal-dual interior point methods to diff-convex problems on symmetric cones

We consider the extension of primal dual interior point methods for linear programming on symmetric cones, to a wider class of problems that includes approximate necessary optimality conditions for functions expressible as the difference of two convex functions of a special form. Our analysis applies the Jordan-algebraic approach to symmetric cones. As the basic method is local, we apply the idea of the filter method for a globalization strategy.     

Complementarity problems over cones with monotone and pseudomonotone maps

The notion of a monotone map is generalized to that of a pseudomonotone map. It is shown that a differentiable, pseudoconvex function is characterized by the pseudomonotonicity of its gradient. Several existence theorems are established for a given complementarity problem over a certain cone where the underlying map is either monotone or pseudomonotone under the assumption that the complementarity problem has a feasible or strictly feasible point.This work was supported in part by the National Science Foundation, Grant No. GP-34619.     

Complex interpolation between two weighted Bergman spaces on tubes over symmetric cones

We prove that the complex interpolation space [A_ν^p₀,A_ν^p₁]_θ, 0<θ<1, between two weighted Bergman spaces A_ν^p₀ and A_ν^p₁ on the tube in $\text{C}{}^{\mathrm{n}}$, n?3, over an irreducible symmetric cone of $\text{R}{}^{\mathrm{n}}$ is the weighted Bergman space A_ν^p with 1/p=(1?θ)/p₀+θ/p₁. Here, ν>n/r?1 and 1?p₀<p₁<2+ν/(n/r?1) where r denotes the rank of the cone. We then construct an analytic family of operators and an atomic decomposition of functions, which are related to this interpolation result. To cite this article: D. Békollé et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Multiobjective symmetric duality involving cones

In this paper, a pair of multiobjective symmetric dual programs over arbitrary cones are formulated for cone-convex functions. Weak, strong, converse and self-duality theorems are proved for these programs.     

Spherical functions on symmetric cones

In this note, we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real Jordan algebra. We use this formula as an inspiration for a similar recursive formula involving the Jack polynomials.