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  • Articles  (45)
  • CR: 5.15  (21)
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  • Articles  (45)
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  • Springer  (45)
  • American Chemical Society
  • American Institute of Physics
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  • 2005-2009
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  • 1
    Electronic Resource
    Electronic Resource
    Conjugate direction methods with variable storage (1982)
    Springer
    Mathematical programming 23 (1982), S. 326-340 
    Details
    ISSN: 1436-4646
    Keywords: Optimization ; Quasi-Newton ; Conjugate Gradient
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we study conjugate gradient algorithms for large optimization problems. These methods accelerate (or precondition) the conjugate gradient method by means of quasi-Newton matrices, and are designed to utilize a variable amount of storage, depending on how much information is retained in the quasi-Newton matrices. We are concerned with the behaviour of such methods on the underlying quadratic model, and in particular, with finite termination properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01583797
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  • 2
    Electronic Resource
    Electronic Resource
    A smoothing-out technique for min—max optimization (1980)
    Springer
    Mathematical programming 19 (1980), S. 61-77 
    Details
    ISSN: 1436-4646
    Keywords: Optimization ; Nonlinear Programming ; Unconstrained Optimization ; Nondifferentiable Optimization ; Min—Max Problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, we suggest approximations for smoothing out the kinks caused by the presence of “max” or “min” operators in many non-smooth optimization problems. We concentrate on the continuous-discrete min—max optimization problem. The new approximations replace the original problem in some neighborhoods of the kink points. These neighborhoods can be made arbitrarily small, thus leaving the original objective function unchanged at almost every point ofR n . Furthermore, the maximal possible difference between the optimal values of the approximate problem and the original one, is determined a priori by fixing the value of a single parameter. The approximations introduced preserve properties such as convexity and continuous differentiability provided that each function composing the original problem has the same properties. This enables the use of efficient gradient techniques in the solution process. Some numerical examples are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581628
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  • 3
    Electronic Resource
    Electronic Resource
    An algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations (1990)
    Springer
    Mathematical programming 47 (1990), S. 175-201 
    Details
    ISSN: 1436-4646
    Keywords: Optimization ; linear programming ; complexity ; polynomial time algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present an algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a precision of O(L) bits.L is bounded by the number of bits in the input. The worst-case running time of the algorithm is better than that of Karmarkar's algorithm by a factor of $$\sqrt {m + n} $$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580859
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  • 4
    Electronic Resource
    Electronic Resource
    Sublinear convergence of the Chord method at singular points (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 147-154 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We give sufficient conditions for convergence of the chord method for a class of singular problems. The rate of convergence is sublinear.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395307
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  • 5
    Electronic Resource
    Electronic Resource
    Error bounds of Newton type process on Banach spaces (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 175-193 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) (1980) ; Primary 65J05, 65J15 ; Secondary 47H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For solving a nonlinear operator equationF(x)=0 in Banach spaces, the Newton's method or Newton type methods are important numerical techniques. We use the properties of real equationt=Φ(t) majorizing an operator equationx=Gx to find a fixed point ofG as a solution of equationF(x)=0. Various type of operatorsG are considered in this paper. For a nonlinear operatorG, we would find a real function Φ majorizing the operatorG and it will be related to a rate of convergence $$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}.$$ It follows thatG has a fixed point as a solution ofF(x)=0. Practical limitations of error bounds like as in Potra and Pták [5] are discribed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408692
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  • 6
    Electronic Resource
    Electronic Resource
    Some convergence results for asynchronous algorithms (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 325-340 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction. We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently. Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.
    Notes: Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction. Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée. Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407866
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  • 7
    Electronic Resource
    Electronic Resource
    Local convergence analysis for partitioned quasi-Newton updates (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 429-448 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers local convergence properties of inexact partitioned quasi-Newton algorithms for the solution of certain non-linear equations and, in particular, the optimization of partially separable objective functions. Using the bounded deterioration principle, one obtains local and linear convergence, which impliesQ-superlinear convergence under the usual conditions on the quasi-Newton updates. For the optimization case, these conditions are shown to be satisfied by any sequence of updates within the convex Broyden class, even if some Hessians are singular at the minimizer. Finally, local andQ-superlinear convergence is established for an inexact partitioned variable metric method under mild assumptions on the initial Hessian approximations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407874
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  • 8
    Electronic Resource
    Electronic Resource
    Bisection is optimal (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 111-117 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We seek an approximation to a zero of a continuous functionf:[a,b]→ℝ such thatf(a)≦0 andf(b)≧0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b−a)/2 n+1, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459080
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  • 9
    Electronic Resource
    Electronic Resource
    On a numerical treatment for the curve-tracing of the homotopy method (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 323-329 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Newton-like iteration scheme is proposed for the tracing of an implicitly defined smooth curve. This scheme originates from the study of the continuous Newton-Raphson method for underdetermined systems and, hence, inherits the characteristic property of orthogonality. Its domain of attraction is formed and makes it possible to trace this curve more efficiently.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389577
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  • 10
    Electronic Resource
    Electronic Resource
    A generalized conjugate gradient algorithm for minimization (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 223-230 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A generalized conjugate gradient algorithm which is invariant to a nonlinear scaling of a strictly convex quadratic function is described, which terminates after at mostn steps when applied to scaled quadratic functionsf: R n →R1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1 (R1) an arbitrary strictly monotone functionh. The algorithm does not suppose the knowledge ofh orF but only off(x) and its gradientg(x).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396318
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  • 11
    Electronic Resource
    Electronic Resource
    On the rate of superlinear convergence of a class of variable metric methods (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 293-313 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 90C30 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers a class of variable metric methods for unconstrained minimization. Without requiring exact line searches each algorithm in this class converges globally and superlinearly on convex functions. Various results on the rate of the superlinear convergence are obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396414
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  • 12
    Electronic Resource
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    Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 93-98 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Zusammenfassung Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.
    Notes: Summary The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459078
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  • 13
    Electronic Resource
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    On the steps of convergence of approximate eigenvectors in the Rayleigh-Schrödinger series (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 31-50 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J10 ; 41A25 ; 41A65 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Error bounds are given for the iterative computation of the eigenvectors in the Rayleigh-Schrödinger series. These bounds remove the discrepancy in the theoretical behaviour and numerical results, noted by Redont, under the assumption of collectively compact convergence. As a particular case, it follows that the eigenvector in the iterative Galerkin method proposed by Sloan improves upon the eigenvector in the Galerkin method. This is illustrated by numerical experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400916
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  • 14
    Electronic Resource
    Electronic Resource
    Imperfect conjugate gradient algorithms for extended quadratic functions (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 97-105 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Generalized conjugate gradient algorithms which are invariant to a nonlinear scaling of a strictly convex quadratic function are described. The algorithms when applied to scaled quadratic functionsf∶R n →R 1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1(R 1) an arbitrary strictly monotone functionh generate the same direction vectors as for the functionF without perfect steps.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400920
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  • 15
    Electronic Resource
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    Sharp error bounds for Newton's process (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 63-72 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01463998
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  • 16
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    Sur la convergence théorique de la méthode du gradient réduit généralisé (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 73-85 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The theoretical convergence of Generalized Reduced Gradient Method (GRG) has not been proved; the purpose of this paper is to propose two theoritical and general variants of the original method and a proof of their convergence.
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    URL: http://dx.doi.org/10.1007/BF01463999
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  • 17
    Electronic Resource
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    The numerical solution of the inverse Stefan problem (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 411-429 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403678
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  • 18
    Electronic Resource
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    Starlike domains of convergence for Newton's method at singularities (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 95-111 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Given a solutionx * of a system of nonlinear equationsf with singular Jacobian ∇f(x *) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox *. Under certain conditions the union of those straight lines throughx *, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396373
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  • 19
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    Some supplementary results on the 1+ $$\sqrt 2 $$ order method for the solution of nonlinear equations (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 383-392 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ $$\sqrt 2 $$ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396439
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  • 20
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    Approximation of optimal distributed control problems governed by variational inequalities (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 393-416 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 49D05, 49D07, 65K10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A method for approximating the optimal control and the optimal state for a class of distributed control problems governed by variational inequalities is given. It uses a Rayleigh-Ritz-Galerkin scheme, regularising techniques and a gradient algorithm. A numerical example is given.
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    URL: http://dx.doi.org/10.1007/BF01396440
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  • 21
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    An error analysis for the secant method (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 427-445 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) 65J15, 65G99 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Sharp a priori and a posteriori error bounds are given for the secant method for solving non-linear equations in Banach spaces. The numerical stability of this method is also investigated. The stability results are analogous to those obtained by Lancaster for Newton's method.
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    URL: http://dx.doi.org/10.1007/BF01396443
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  • 22
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    Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 455-465 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10, 47H17 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A numerical technique is described for computing turning points of a space curveL implicitly defined by a nonlinear system ofn equations inn+1 variables. The basic idea is a local parametrization ofL where the parameter that gives the next iterate is determined by applying one step of the well-known method for minimizing a real function using cubic Hermite interpolation with two nodes. The method is shown to convergeQ-super-linearly and withR-order of at least two. A numerical example concerning the analysis of nonlinear resistive circuits shows the algorithm to work effectively on real life problems.
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    URL: http://dx.doi.org/10.1007/BF01396445
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  • 23
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    A generalization of regula falsi (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 333-346 
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    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to a multistep variant of the secant method. Optimal conditions for convergence as well as error estimates, sharp in every step, are obtained.
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    URL: http://dx.doi.org/10.1007/BF01396659
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  • 24
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    Partitioned variable metric updates for large structured optimization problems (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 119-137 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents a minimization method based on the idea of partitioned updating of the Hessian matrix in the case where the objective function can be decomposed in a sum of convex “element” functions. This situation occurs in a large class of practical problems including nonlinear finite elements calculations. Some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented.
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    URL: http://dx.doi.org/10.1007/BF01399316
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  • 25
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    A new augmented Lagrangian function for inequality constraints in nonlinear programming problems (1982)
    Springer
    Journal of optimization theory and applications 36 (1982), S. 495-519 
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    ISSN: 1573-2878
    Keywords: Optimization ; nonlinear programming ; Numerical methods ; computational methods ; augmented Lagrangian functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.
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    URL: http://dx.doi.org/10.1007/BF00940544
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  • 26
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    Equivalence theorem for constrained optimization of the LegendreK-transform of a concave function (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 25-31 
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    ISSN: 1573-2878
    Keywords: Optimization ; Kuhn-Tucker theorem ; LegendreK-transform ; equilibrium composition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The most common way of determining the steady states of a system is through the optimization of a concave function on a convex set. This applies only to cases where the objective function depends exclusively onextensive variables. In fields such as econometrics, physics, or chemistry, controllable quantities (and therefore constrained variables) are generally referred to asintensive parameters, and the states are described through a potential function. In the following pages, we examine how these two aspects can be connected.
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    URL: http://dx.doi.org/10.1007/BF00934320
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  • 27
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    An approach to the solution of the vector optimization problem of dynamic systems (1982)
    Springer
    Journal of optimization theory and applications 38 (1982), S. 409-422 
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    ISSN: 1573-2878
    Keywords: Optimization ; scalar optimization ; vector optimization ; optimization under vector-valued criteria ; maximum principle ; dynamic programming ; optimization of dynamic systems ; multi-criteria decision problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control problem with vector-valued criteria is considered. A new approach to the generalization of this problem and a method of constructing the Bellman function are given.
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  • 28
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    Existence of value in differential games with terminal cost function (1983)
    Springer
    Journal of optimization theory and applications 39 (1983), S. 89-104 
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    ISSN: 1573-2878
    Keywords: Differential games ; optimal strategies ; value of a game ; terminal cost function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A two-person, zero-sum differential game with general type phase constraints and terminal (not fixed) cost function is investigated. Player II (possessing complete information) can choose any strategy in the Varaiya-Lin sense, while his opponent (having incomplete information) can select any lower II-strategy introduced by Friedman (Ref. 1). The existence of a value and an optimal player II's strategy is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. The present paper largely extends the results of Ref. 2.
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    URL: http://dx.doi.org/10.1007/BF00934607
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    Multimodel design of a Nash strategy (1980)
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    Journal of optimization theory and applications 31 (1980), S. 553-564 
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    ISSN: 1573-2878
    Keywords: Differential games ; perturbation theory ; systems order reduction ; large-scale systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For interconnected systems with slow and fast modes, we analyze a multimodel design of a Nash equilibrium strategy. Decision makers use different models of the same system to obtain a composite strategy that approximates the exact Nash strategy. It is discussed in what sense this approximation holds.
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    URL: http://dx.doi.org/10.1007/BF00934477
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    Remarks on capture-avoidance games (1980)
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    Journal of optimization theory and applications 32 (1980), S. 365-377 
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    ISSN: 1573-2878
    Keywords: Differential games ; pursuit and evasion ; capture avoidance ; linear games
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    Topics: Mathematics
    Notes: Abstract We formulate the problems of capture and avoidance as differential games with terminal cost, thus avoiding some difficulties arising from choosing the terminal time or the closest distance as the cost. This puts capture and avoidance in the context of differential games, in the sense of Isaacs. The results are illustrated via linear differential games with hard control constraints.
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    System theory: A new approach to wave mechanics (1980)
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    Journal of optimization theory and applications 32 (1980), S. 463-478 
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    ISSN: 1573-2878
    Keywords: Optimization ; control ; relativity ; stochastic processes ; wave mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The intention of this paper is to show how the influence of the work of Bellman has initiated a tentative new approach to quantum mechanics
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    URL: http://dx.doi.org/10.1007/BF00934034
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    Shape design sensitivity of a membrane (1983)
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    Journal of optimization theory and applications 40 (1983), S. 595-623 
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    ISSN: 1573-2878
    Keywords: Optimization ; partial differential equations ; elasticity ; control ; design sensitivity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dependence of the static response and the eigenvalues of a membrane on its shape is characterized. A transformation function is defined to determine the shape of the membrane. Differential operator properties and transformation techniques of integral calculus are employed to show that the static response and the eigenvalues of the system depend in a continuous and differentiable way on the shape of the membrane. Explicit and computable formulas are presented for the derivative (first variation) of the structural response and the eigenvalues with respect to the shape. A rigorous proof is provided, and the shape design sensitivity of a typical integral functional is determined.
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    On qualitative differential games with two targets (1983)
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    Journal of optimization theory and applications 41 (1983), S. 587-598 
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    ISSN: 1573-2878
    Keywords: Differential games ; first reachability of sets ; playability ; pursuit-evasion problem
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    Topics: Mathematics
    Notes: Abstract We consider a qualitative differential game between two players, each with their own respective target. A player's objective is to steer the response of a dynamical system under the control of both players into his own target before his opponent can steer it to his. The sufficiency conditions given by Getz and Leitmann (Ref. 2), which ensure termination on a particular target, are improved. The theory is illustrated by a pursuit-evasion problem.
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    On the minimality of search paths under absolutely continuous univariate distributions (1990)
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    Mathematical methods of operations research 34 (1990), S. 43-58 
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    ISSN: 1432-5217
    Keywords: Linear Search ; Optimization ; Absolutely Continuous distributions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Ein Objekt, das sich in zufälliger Position auf einer Strecke befindet, soll gesucht werden; der Startpunkt liegt im Inneren der Strecke. Für die Bewegungsgeschwindigkeit bei der Suche ist eine obere Schranke gegeben; der Suchpfad muß stetig sein, wobei beliebig viele Richtungsänderungen zugelassen sind. Das Ziel ist die Minimierung der erwarteten Zeit bis zum Erreichen des Objekts. Unter der Voraussetzung einer absolut stetigen Verteilung für die Position des Objekts werden hinreichende Bedingungen für die Optimalität eines Suchpfades angegeben.
    Notes: Abstract The linear search problem is concerned with the search for a target located randomly on a line. The target being sought for might be on either side of some starting point. A searcher makes for the target with an upper bound on his speed, using a continuous search path in which he changes his direction many times before reaching his goal. The minimality of average distance (time) from the starting point upon reaching the target is the measure of optimality of search paths. We have obtained sufficient conditions that guarantee the minimality of such search paths whenever the target has an absolutely continuous distribution.
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    On the isochronous and stroboscopic capture in differential games (1982)
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    Journal of optimization theory and applications 37 (1982), S. 115-119 
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    ISSN: 1573-2878
    Keywords: Differential games ; isochronous capture ; stroboscopic strategy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Ref. 1, Hájek formulated the conjecture that isochronous capture is necessarily stroboscopic in linear differential games. Here, it is shown that this conjecture is not valid.
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    URL: http://dx.doi.org/10.1007/BF00934369
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    Multiplier rules and the separation of convex sets (1983)
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    Journal of optimization theory and applications 40 (1983), S. 321-331 
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    ISSN: 1573-2878
    Keywords: Optimization ; necessary conditions ; Kuhn-Tucker theorem ; multiplier rules ; separation of convex sets ; mathematical programming ; generalized derivatives
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the following abstract mathematical programming problem: in a setD, find an element that optimizes a real function φ0, subject to inequality constraints φ1⩽0, ..., φ p ⩽0 and equality constraints φ p+1=0, ..., φ p+q =0. Necessary conditions for this problem, like the Karush-Kuhn-Tucker theorem, can be seen as a consequence of separating with a hyperplane two convex sets inR p+q+1, the image space of the map Φ=(φ0, φ1, ..., φ p+q ). This paper reviews this approach and organizes it into a coherent way of looking at necessary conditions in optimization theory.
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    A class of differential games for which the closed-loop and open-loop Nash equilibria coincide (1982)
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    Journal of optimization theory and applications 36 (1982), S. 253-262 
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    ISSN: 1573-2878
    Keywords: Differential games ; Nash equilibria ; open-loop strategies ; closed-loop strategies
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is well known that, in general, Nash equilibria in open-loop strategies do not coincide with those in closed-loop strategies. This note identifies a class of differential games in which the Nash equilibrium in closed-loop strategies is degenerate, in the sense that it depends on time only. Consequently, the closed-loop equilibrium is also an equilibrium in open-loop strategies.
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    URL: http://dx.doi.org/10.1007/BF00933832
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    On the convergence, invariance, and related aspects of a modification of Huang's algorithm (1982)
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    Journal of optimization theory and applications 37 (1982), S. 315-341 
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    ISSN: 1573-2878
    Keywords: Optimization ; unconstrained minimization ; updates ; line searches ; convergence ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider a certain generalization of the Huang family of updates and discuss, firstly, convergence, dependence on parameters, and descent property; secondly, invariance under nonlinear scaling, conjugacy of search directions, and possibility of achieving a better approximation of the inverse of the Hessian. The last three aspects are shown to be dependent on particular choices of parameters. A numerical experiment is presented comparing the performances of the usual and modified BFGS algorithms.
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    URL: http://dx.doi.org/10.1007/BF00935273
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    Differential game of optimal approach of two inertial pursuers to a noninertial evader (1990)
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    Journal of optimization theory and applications 65 (1990), S. 501-518 
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    ISSN: 1573-2878
    Keywords: Differential games ; value function ; programmed maximin function ; dispersal surface ; focal surface
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the game of the optimal approach of two identical inertial pursuers to a noninertial evader is investigated. The duration of the game is fixed. The payoff functional is the distance between the evader and the closest pursuer when the game terminates. The value function is constructed for all possible positions of the game. The regions where the pursuit is one-to-one and the regions where it is essentially collective are described algorithmically. Some analogies between this game and the linear differential game with elliptical vectograms are indicated. It is noted that the focal surface and the dispersal surface are in proximity of one another.
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    Singular differential game numerical techniques (1982)
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    Journal of optimization theory and applications 37 (1982), S. 69-87 
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    ISSN: 1573-2878
    Keywords: Differential games ; singular arcs ; saddle points ; two-point boundary-value problems ; switching times ; closed-loop solutions ; inverse systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper develops a numerical technique to solve a class of zero-sum differential games with singular control. By using this technique and the application of inverse systems, a near-optimal closed-loop technique is developed to generate a numerical solution to this class of problems.
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    URL: http://dx.doi.org/10.1007/BF00934367
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    Remarks on capture-avoidance games (1982)
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    Journal of optimization theory and applications 37 (1982), S. 121-122 
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    ISSN: 1573-2878
    Keywords: Differential games ; pursuit and evasion ; captureavoidance ; linear games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Corrections to some results of Ref. 1 are stated.
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    Existence of value in differential games with fixed time duration (1982)
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    Journal of optimization theory and applications 38 (1982), S. 581-598 
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    ISSN: 1573-2878
    Keywords: Differential games ; optimal strategies ; value of a game
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A differential game of prescribed duration with general-type phase constraints is investigated. The existence of a value in the Varaiya-Lin sense and an optimal strategy for one of the players is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. These results are next widened on more general games, examined earlier by Varaiya.
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    Feedback Nash equilibria in a problem of optimal fishery management (1990)
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    Journal of optimization theory and applications 64 (1990), S. 293-310 
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    ISSN: 1573-2878
    Keywords: Differential games ; feedback Nash equilibria ; renewable resources ; common-property fishery
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper deals with a problem of optimal management of a common-property fishery, modelled as a two-player differential game. Under nonclassical assumptions on harvest rates and utilities, a feedback Nash equilibrium is determined, using a bionomic equilibrium concept. Later on, this assumption is relaxed and a feedback Nash equilibrium is established under minimal hypotheses.
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    Optimization technique based on learning automata (1990)
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    Journal of optimization theory and applications 64 (1990), S. 331-347 
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    ISSN: 1573-2878
    Keywords: Optimization ; learning systems ; nonconvex programming ; process synthesis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Optimization techniques are finding increasingly numerous applications in process design, in parallel to the increase of computer sophistication. The process synthesis problem can be stated as a largescale constrained optimization problem involving numerous local optima and presenting a nonlinear and nonconvex character. To solve this kind of problem, the classical optimization methods can lead to analytical and numerical difficulties. This paper describes the feasibility of an optimization technique based on learning systems which can take into consideration all the prior information concerning the process to be optimized and improve their behavior with time. This information generally occurs in a very complex analytical, empirical, or know-how form. Computer simulations related to chemical engineering problems (benzene chlorination, distillation sequence) and numerical examples are presented. The results illustrate both the performance and the implementation simplicity of this method.
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    Optimization of control setting and size of shunt capacitors on distribution feeders (1990)
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    Journal of optimization theory and applications 65 (1990), S. 305-319 
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    ISSN: 1573-2878
    Keywords: Optimization ; shunt capacitors ; distribution feeders ; control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Loads on electric utility systems have two components: active power (measured in kilowatts) and reactive power (measured in kilovars). Active power has to be generated at the power plant, whereas reactive power can be provided by either power plants or capacitors. It is a well-known fact that shunt power capacitors are the most economical source to meet the reactive power requirements of inductive loads and transmission lines operating at a lagging power factor. This paper describes new contributions to the problem of optimization of size and control setting of shunt capacitors on distribution feeders, so that the losses along the feeder are minimized. The variation of the KVAR of the load on the feeder with the distance from the substation is assumed to be linear. The parameters of this function are estimated from the available KVAR loading on the feeder first by using least-square techniques and then by using least-absolute-value parameter estimation techniques. The results obtained are compared with that obtained if the current profile is assumed to be uniformly distributed on the feeder. Our results show that the optimum size of the capacitor bank as well as its optimum location depend on the parameters of the model used for the load. Also, our results show a large saving in the size of the capacitor banks used with a considerable per-unit-loss reduction along the feeder.
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    URL: http://dx.doi.org/10.1007/BF01102349
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