First order gradient optimization in LISP

In this paper we develop algorithms in programming language SCHEME for implementation of the main first order gradient techniques for unconstrained optimization. Implementation of the descent techniques which use non-optimal descent steps as well as implementation of the optimal descent techniques are described. Also, we investigate implementation of the global problem, called optimization along a line. Developed programs are effective and simpler with respect to the corresponding in the procedural programming languages. Several numerical examples are reported.     

First and second order optimality conditions for set-valued optimization problems

In this paper we study first and second order necessary and sufficient optimality conditions for optimization problems involving set-valued maps and we derive some known results in a more general framework.     

Optimality and duality for multiple-objective optimization under generalized type I univexity

In this paper, we extend the classes of generalized type I vector-valued functions introduced by Aghezzaf and Hachimi in [J. Global Optim. 18 (2000) 91-101] to generalized univex type I vector-valued functions and consider a multiple-objective optimization problem involving generalized type I univex functions. A number of Kuhn-Tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution. The Mond-Weir and general Mond-Weir type duality results are also presented.     

First and second order conditions for a class of nondifferentiable optimization problems

This paper gives first and in particular second order necessary and sufficient conditions for a class of nondifferentiable optimization problems in which there are both objective and constraint functions defined in terms of a norm. The conditions are expressed in terms of a Lagrangian function and its derivatives, and use the ideas of feasible directional sequence and subgradients. Certain regularity assumptions are required and for the second order necessary conditions it is shown that the assumption is realistic for polyhedral norms. Illustrative examples are discussed.     

Optimality criteria and duality in multiple-objective optimization involving generalized invexity

A multiple-objective optimization problem involving generalized invex functions is considered. Kuhn-Tucker type necessary and sufficient conditions are obtained for a feasible point to be an efficient or properly efficient solution. Two dual programs are obtained. The results are given under weaker invexity assumptions.     

First order optimality conditions in vector optimization involving stable functions

We study a nonsmooth vector optimization problem with an arbitrary feasible set or a feasible set defined by a generalized inequality constraint and an equality constraint. We assume that the involved functions are nondifferentiable. First, we provide some calculus rules for the contingent derivative in which the stability (a local Lipschitz property at a point) of the functions plays a crucial role. Second, another calculus rules are established for steady functions. Third, necessary optimality conditions are stated using tangent cones to the feasible set and the contingent derivative of the objective function. Finally, some necessary and sufficient conditions are presented through Lagrange multiplier rules.     

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.     

First order perturbation theory for harmonic oscillators

Zusammenfassung Eine Störungsrechnung erster Ordnung wird für ein autonomes System von gestörten harmonischen Oszillatoren hergeleitet mittels einer Verallgemeinerung der Methode von Krylow-Bogoljubow.     

First and second order optimality conditions in vector optimization problems with nontransitive preference relation

We present first and second order conditions, both necessary and sufficient, for ?-minimizers of vector-valued mappings over feasible sets with respect to a nontransitive preference relation ?. Using an analytical representation of a preference relation ? in terms of a suitable family of sublinear functions, we reduce the vector optimization problem under study to a scalar inequality, from which, using the tools of variational analysis, we derive minimality conditions for the initial vector optimization problem.     

Optimal multiple-objective resource allocation using hybrid particle swarm optimization and adaptive resource bounds technique

The multiple-objective resource allocation problem (MORAP) seeks for an allocation of resource to a number of activities such that a set of objectives are optimized simultaneously and the resource constraints are satisfied. MORAP has many applications, such as resource distribution, project budgeting, software testing, health care resource allocation, etc. This paper addresses the nonlinear MORAP with integer decision variable constraint. To guarantee that all the resource constraints are satisfied, we devise an adaptive-resource-bound technique to construct feasible solutions. The proposed method employs the particle swarm optimization (PSO) paradigm and presents a hybrid execution plan which embeds a hill-climbing heuristic into the PSO for expediting the convergence. To cope with the optimization problem with multiple objectives, we evaluate the candidate solutions based on dominance relationship and a score function. Experimental results manifest that the hybrid PSO derives solution sets which are very close to the exact Pareto sets. The proposed method also outperforms several representatives of the state-of-the-art algorithms on a simulation data set of the MORAP.     

First order autoregressive markov processes

This paper is concerned with establishing conditions under which finite (and then countably infinite) stationary Markov chains have first order autoregressive representations.     

First order convergence of matroids

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini–Schramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branch-depth assumption nor the representability assumption can be removed.     

On Natural First order Lagrangians

The natural first order Lagrangian is a function defined onpairs of Riemannian manifolds and which is invariant with respectto immersions and depends continuously on 1-jets of metrics.We prove that there is a canonical bijection between naturalfirst order Lagrangians and functions which are smooth, symmetricand even.     

Third order efficiency of conditional tests for exponential families

Let P_η, η = (θ, γ) ∈ Θ × Γ ? $\text{R}$ × $\text{R}$^k, be a (k + 1)-dimensional exponential family. Let $\text{?}{}_{\mathrm{n}}{}^1$, n ∈ $\text{N}$, be an optimal similar test for the hypothesis {P_(θ,γ)ⁿ: γ ∈ Γ} (θ ∈ Θ fixed) against alternatives P_(θ1,γ1)ⁿ, θ₁ > θ, γ₁ ∈ Γ. It is shown that (?_n¹)_{n∈$\text{N}$} is third order efficient in the class of all test-sequences that are asymptotically similar of level α + o(n^?1) (locally uniformly in the nuisance parameter γ).     

First order spectrum for elliptic system and nonresonance problem

In this paper we will study the first order spectrum for elliptic systems and the existence of solutions for a quasilinear elliptic system under the condition of nonresonance below the first eigensurface. This revised version was published online in June 2006 with corrections to the Cover Date.     

An interactive method for multiple-objective mathematical programming problems

An interactive method is developed for solving the general nonlinear multiple objective mathematical programming problems. The method asks the decision maker to provide partial information (local tradeoff ratios) about his utility (preference) function at each iteration. Using the information, the method generates an efficient solution and presents it to the decision maker. In so doing, the best compromise solution is sought in a finite number of iterations. This method differs from the existing feasible direction methods in that (i) it allows the decision maker to consider only efficient solutions throughout, (ii) the requirement of line search is optional, and (iii) it solves the problems with linear objective functions and linear utility function in one iteration. Using various problems selected from the literature, five line search variations of the method are tested and compared to one another. The nonexisting decision maker is simulated using three different recognition levels, and their impact on the method is also investigated.     

First order properties of random posets

Let =(n,p) be a binary relation on the set [n]={1, 2, ..., n} such that (i,i) for every i and (i,j) with probability p, independently for each pair i,j [n], where i<j. Define as the transitive closure of and denote poset ([n], ) by R(n, p). We show that for any constant p probability of each first order property of R(n, p) converges as n .     

First order theory of permutation groups

We solve the problem of the elementary equivalence (definability) of the permutation groups over cardinals ℵ_α. We show that it suffices to solve the problem of elementary equivalence (definability) for the ordinals α in certain second order logic, and this is reduced to the case of α < (2^ℵ ₀)⁺. We solve a problem of Mycielski and McKenzie on embedding of free groups in permutation groups, and discuss some weak second-order quantifiers. An erratum to this article is available at .