Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.     

On the Second-order Directional Derivatives of Singular Values of Matrices and Symmetric Matrix-valued Functions

The (parabolic) second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions induced by real-valued functions play important roles in studying second-order optimality conditions for different types of matrix cone optimization problems. We propose a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a symmetric matrix in Torki (Nonlinear Anal 46:1133–1150 2001), from which a formula for the second-order directional derivative of any singular value of a matrix is established. We demonstrate a formula for the second-order directional derivative of the symmetric matrix-valued function. As applications, the second-order derivative for the projection operator over the SDP cone is derived and used to get the second-order tangent set of the SDP cone in Bonnans and Shapiro (2000), and the tangent cone and the second-order tangent set of the epigraph of the nuclear norm are given as well.     

Parabolic Second-Order Directional Differentiability in the Hadamard Sense of the Vector-Valued Functions Associated with Circular Cones

In this paper, we study the parabolic second-order directional derivative in the Hadamard sense of a vector-valued function associated with circular cone. The vector-valued function comes from applying a given real-valued function to the spectral decomposition associated with circular cone. In particular, we present the exact formula of second-order tangent set of circular cone by using the parabolic second-order directional derivative of projection operator. In addition, we also deal with the relationship of second-order differentiability between the vector-valued function and the given real-valued function. The results in this paper build fundamental bricks to the characterizations of second-order necessary and sufficient conditions for circular cone optimization problems.     

Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

     

Generalized second-order directional derivatives and optimization with C1,1 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C¹,¹ functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C^1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.     

Second-order sufficient conditions for control problems with mixed control-state constraints

References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.The authors wish to thank K. Malanowski for helpful discussions.     

Multiple periodic solutions for a second-order problem on periodic time scales

Green's function for a second-order delta dynamic equation is found, then used as the kernel in an integral operator to guarantee the existence of positive periodic solutions to a second-order periodic boundary value problem with periodic coefficient functions.     

A one-step smoothing Newton method for second-order cone programming

A new smoothing function for the second-order cone programming is given by smoothing the symmetric perturbed Fischer–Burmeister function. Based on this new function, a one-step smoothing Newton method is presented for solving the second-order cone programming. The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. This algorithm does not have restrictions regarding its starting point and is Q-quadratically convergent. Numerical results suggest the effectiveness of our algorithm.     

On the No-Gap Second-Order Optimality Conditions for a Discrete Optimal Control Problem with Mixed Constraints

Motivated by our recent works on optimality conditions in discrete optimal control problems under a nonconvex cost function, in this paper, we study second-order necessary and sufficient optimality conditions for a discrete optimal control problem with a nonconvex cost function and state-control constraints. By establishing an abstract result on second-order optimality conditions for a mathematical programming problem, we derive second-order necessary and sufficient optimality conditions for a discrete optimal control problem. Using a common critical cone for both the second-order necessary and sufficient optimality conditions, we obtain “no-gap” between second-order optimality conditions.     

An unconstrained smooth minimization reformulation of the second-order cone complementarity problem

A popular approach to solving the nonlinear complementarity problem (NCP) is to reformulate it as the global minimization of a certain merit function over ℝⁿ. A popular choice of the merit function is the squared norm of the Fischer-Burmeister function, shown to be smooth over ℝⁿ and, for monotone NCP, each stationary point is a solution of the NCP. This merit function and its analysis were subsequently extended to the semidefinite complementarity problem (SDCP), although only differentiability, not continuous differentiability, was established. In this paper, we extend this merit function and its analysis, including continuous differentiability, to the second-order cone complementarity problem (SOCCP). Although SOCCP is reducible to a SDCP, the reduction does not allow for easy translation of the analysis from SDCP to SOCCP. Instead, our analysis exploits properties of the Jordan product and spectral factorization associated with the second-order cone. We also report preliminary numerical experience with solving DIMACS second-order cone programs using a limited-memory BFGS method to minimize the merit function. In honor of Terry Rockafellar on his 70th birthday     

On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit L _p -viscosity solutions, which are in C ^1+α for an ${\alpha \in (0, 1)}$ . The equations have a special structure that the “main” part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.     

Complément de Schur et sous-différentiel du second ordre d'une fonction convexe

Summary The Schur complement relative to the linear mappingA of a functionf is denotedAf and defined as the image off underA. In this paper we give some estimates for the second-order differential ofAf whenf is either a partially quadratic convex function or aC ² convex function with a nonsingular second-order differential. We then consider an arbitrary convex functionf and study the second-order differentiability ofAf in a more general sense.

     

Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets

We provide second-order necessary and sufficient conditions for a point to be an efficient element of a set with respect to a cone in a normed space, so that there is only a small gap between necessary and sufficient conditions. To this aim, we use the common second-order tangent set and the asymptotic second-order cone utilized by Penot. As an application we establish second-order necessary conditions for a point to be a solution of a vector optimization problem with an arbitrary feasible set and a twice Fréchet differentiable objective function between two normed spaces. We also establish second-order sufficient conditions when the initial space is finite-dimensional so that there is no gap with necessary conditions. Lagrange multiplier rules are also given. This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), Project BFM2003-02194. Online publication 29 January 2004.     

A descent method for a reformulation of the second-order cone complementarity problem

Analogous to the nonlinear complementarity problem and the semi-definite complementarity problem, a popular approach to solving the second-order cone complementarity problem (SOCCP) is to reformulate it as an unconstrained minimization of a certain merit function over Rⁿ${\mathbb{R}}^n$. In this paper, we present a descent method for solving the unconstrained minimization reformulation of the SOCCP which is based on the Fischer–Burmeister merit function (FBMF) associated with second-order cone [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327], and prove its global convergence. Particularly, we compare the numerical performance of the method for the symmetric affine SOCCP generated randomly with the FBMF approach [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327]. The comparison results indicate that, if a scaling strategy is imposed on the test problem, the descent method proposed is comparable with the merit function approach in the CPU time for solving test problems although the former may require more function evaluations.     

Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.     

Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACT

By utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.     

The penalized Fischer-Burmeister SOC complementarity function

In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone complementarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian R ₀₁-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.     

Trigonometric polynomial or exponential fitting approach?

The application of a trigonometric polynomial and an exponential fitting approach is compared for a three-point formula for second-order derivatives, for Simpson’s quadrature rule and for Numerov’s scheme for second-order differential equations. The expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied. The errors for specific problems obtained in both the approaches as a function of the frequency are compared.     

Second-Order Differential and Sensitivity Properties of Weak Vector Variational Inequalities

In this paper, by using the second-order contingent derivative, second-order differential properties of a class of set-valued maps are investigated and an explicit expression of the second-order contingent derivatives is obtained. Then, by means of a gap function, second-order sensitivity properties are discussed for a weak vector variational inequality.