Minimization of vectors of curvilinear functionals on the second order jet bundle

Consider the multi-time multi-objective variational problem (MFP) of minimizing a vector of quotients of path independent curvilinear functionals subject to PDE and/or PDI constraints. The goal of our work is to develop an optimization theory for the second order jet bundle. While the background in Sect. 1 is introductory, the theory in Sects. 2 and 3 is new as a whole, containing our results.     

On sufficient second order optimality conditions in multiobjective optimization

A second order sufficient optimality criterion is presented for a multiobjective problem subject to a constraint given just as a set. To this aim, we first refine known necessary conditions in such a way that the sufficient ones differ by the replacement of inequalities by strict inequalities. Furthermore, we show that no relationship holds between this criterion and a sufficient multipliers rule, when the constraint is described by inequalities and equalities. Finally, improvements of this criterion for the unconstrained case are presented, stressing the differences with single-objective optimization     

Necessary and sufficient conditions for oscillation of second order linear differential equations

Necessary and sufficient conditions are given for oscillations of second order linear differential equationx″+p(t)x′+q(t)x=0.     

First and second order sufficient conditions for optimal control and the calculus of variations

In this paper we develop first and second order sufficient conditions for optimal control and the calculus of variations problems. Our conditions are derived from the Hamilton-Jacobi approach [15, Thm. 2], which was obtained for the generalized problem of Bolza. We do not require any convexity on the data [7] and [11], or that the control setU is polyhedral [14], or that the control function is in the interior ofU [8]. Instead, we assume a certain inequality which is satisfied in each of the above mentioned cases.The publication of this report has been made possible due to a grant of the Fonds FCAC for the help and support of research.     

First and second order sufficient conditions for strict minimality in nonsmooth vector optimization

In this paper we present first and second order sufficient conditions for strict local minima of orders 1 and 2 to vector optimization problems with an arbitrary feasible set and a twice directionally differentiable objective function. With this aim, the notion of support function to a vector problem is introduced, in such a way that the scalar case and the multiobjective case, in particular, are contained. The obtained results extend the multiobjective ones to this case. Moreover, specializing to a feasible set defined by equality, inequality, and set constraints, first and second order sufficient conditions by means of Lagrange multiplier rules are established.     

Necessary and sufficient second order optimality conditions for multiobjective problems with C1 data

In this paper, we obtain necessary and sufficient second order optimality conditions for multiobjective problems using second order directional derivatives. We propose the notion of second order KT-pseudoinvex problems and we prove that this class of problems has the following property: a problem is second order KT-pseudoinvex if and only if all its points that satisfy the second order necessary optimality condition are weakly efficient. Also we obtain second order sufficient conditions for efficiency.     

Minimization of integral functionals depending on lipschitz domains

In this paper is solved a minimization problem for what is essentially an integral functional depending on domains which verify an uniform cone property with a fixed parameter θ by extending the techniques land results of O. Caligaris and P. Oliva ‘1’ for convex sets. A Dirichlet condition and an obstacle are considered.     

Necessary and sufficient condition on global optimality without convexity and second order differentiability

The main goal of this paper is to give a necessary and sufficient condition of global optimality for unconstrained optimization problems, when the objective function is not necessarily convex. We use Gâteaux differentiability of the objective function and its bidual (the latter is known from convex analysis).     

Necessary and sufficient conditions for a minimum of mixed order

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 415–419, March, 1989.     

Infinitesimal affine transformations of the second order tangent bundle with horizontal lift connection

We obtain the canonical expansion of an infinitesimal affine transformation of the second order tangent bundle with horizontal lift connection. We establish necessary and sufficient conditions under which a vector field is an infinitesimal affine transformation. We also construct the horizontal lift of a linear connection to a second order Weil bundle.     

On the limit laws of the second order for additive functionals of Harris recurrent Markov chains

Let {X _n } _{n ≥0} be a Harris recurrent Markov chain with state space E and invariant measure π. The law of the iterated logarithm and the law of weak convergence are given for the additive functionals of the form

Minimization of functionals on the solution of a large-scale discrete ill-posed problem

In this work we study the minimization of a linear functional defined on a set of approximate solutions of a discrete ill-posed problem. The primary application of interest is the computation of confidence intervals for components of the solution of such a problem. We exploit the technique introduced by Eldén in 1990, utilizing a parametric programming reformulation involving the solution of a sequence of quadratically constrained least squares problems. Our iterative method, which uses the connection between Lanczos bidiagonalization and Gauss-type quadrature rules to bound certain matrix functionals, is well-suited for large-scale problems, and offers a significant reduction in matrix-vector product evaluations relative to available methods.     

On the lower bound in second order estimation for Poisson processes: Asymptotic efficiency

In the estimation problem of the mean function of an inhomogeneous Poisson process there is a class of kernel type estimators that are asymptotically efficient alongside with the empirical mean function. We start by describing such a class of estimators which we call first order efficient estimators. To choose the best one among them we prove a lower bound that compares the second order term of the mean integrated square error of all estimators. The proof is carried out under the assumption on the mean function Λ(·) that Λ(τ) = S, where S is a known positive number. In the end, we discuss the possibility of the construction of an estimator which attains this lower bound, thus, is asymptotically second order efficient.     

Continuous linear functionals on certain function spaces satisfying a system of two singular second order differential equations

In this paper, we study a system of two singular second order differential equations, which arises from the theory of harmonic analysis on complex symmetric spaces. First of all, the distributional solutions on an neighborhood of zero in R² are determined. Next, some new function spaces are introduced and the system is solved in the duals of these new spaces.     

任意阶中立型方程正解存在的充要条件

给出了任意阶中立型微分方程（x(t)-p(t)g(x(τ(t)))^(n) ∫α^βt(t,ξ，x(g1(t,ξ）），x(g2(t,ξ）），…，x(gm(t,ξ）））dη（ξ）=0存在正解x(t)满足x(t)-p(t)g(x(τ(t)))/t^k→正常数（→∞)物条件，作为本文结果的特例，部分地解决了文[5]提出的公开问题2。     

Minimization of convex functionals involving nested maxima: Nonconcave duality and algorithms

We consider problems of the form $$\mathop {\min }\limits_u J(u) + \sum\limits_{0 \leqslant i \leqslant n} {a_i } \mathop {\max }\limits_{0 \leqslant j \leqslant i} \Theta _j (u),$$ in which thea′ _i s are nonnegative numbers, andJ and the Θ _j 's are convex functionals on a reflexive Banach spaceU. We show that such problems may arise, in particular, when scheduling investments over several periods of time. By expressing the nested maximizations in a recursive way, we transform the above problem into that of finding the minimax of some functional Φ(u, α), where α ranges over the unit cube of ? ⁿ . Although the dependence of Φ on α is neither linear nor even concave, we show that a saddle point does exist for this problem. Moreover, we propose a very simple dual algorithm to solve max_α min _u Φ(u, α), whose convergence is proved and whose limit yields the true optimum, although the dual functional is not concave and does have local maxima in general. Another algorithm with this same property, but without convergence proof, is also proposed. Finally, a numerical example illustrates the use of these approaches and algorithms.     

A linear programming proof of the second order conditions of non-linear programming

In this note we give a new, simple proof of the standard first and second order necessary conditions, under the Mangasarian–Fromovitz constraint qualification (MFCQ), for non-linear programming problems. We work under a mild constraint qualification, which is implied by MFCQ. This makes it possible to reduce the proof to the relatively easy case of inequality constraints only under MFCQ. This reduction makes use of relaxation of inequality constraints and it makes use of a penalty function. The new proof is based on the duality theorem for linear programming; the proofs in the literature are based on results of mathematical analysis. This paper completes the work in a recent note of Birbil et al. where a linear programming proof of the first order necessary conditions has been given, using relaxation of equality constraints.     

Abstract quasilinear equations of second order with Wentzell boundary conditions

We introduce a general framework to treat abstract quasilinear equations of second order with Wentzell boundary conditions. As an example we study a wave equation for a second order quasilinear differential operator on with Wentzell boundary conditions.