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  • Articles  (18)
  • Mathematical programming  (18)
  • 1975-1979  (18)
  • Mathematics  (18)
  • Energy, Environment Protection, Nuclear Power Engineering
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  • Articles  (18)
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  • Mathematics  (18)
  • Energy, Environment Protection, Nuclear Power Engineering
  • 1
    Electronic Resource
    Electronic Resource
    On the rate of convergence of some feasible direction algorithms (1976)
    Springer
    Journal of optimization theory and applications 20 (1976), S. 297-313 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; inequality constraints ; numerical methods ; descent methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with first-order methods of feasible directions. Pironneau and Polak have recently proved theorems which show that three of these methods have a linear rate of convergence for certain convex problems in which the objective functions have positive definite Hessians near the solutions. In the present note, it is shown that these theorems on rate of convergence can be extended to larger classes of problems. These larger classes are determined in part by certain second-order sufficiency conditions, and they include many nonconvex problems. The arguments used here are based on the finite-dimensional version of Hestenes' indirect sufficiency method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933625
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  • 2
    Electronic Resource
    Electronic Resource
    Generalized reduced gradient method as an extension of feasible direction methods (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 209-226 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; gradient methods ; feasible direction methods ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper presents modifications of the generalized reduced gradient method which allows for a convergence proof. For that, a special construction of the basis is introduced, and some tools of the theory of feasible direction are used to modify the common choice of the direction at every step.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933163
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  • 3
    Electronic Resource
    Electronic Resource
    On a characterization of convergence for the Hestenes method of multipliers (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 227-237 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; method of multipliers ; necessary conditions ; sufficient conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The connection between the convergence of the Hestenes method of multipliers and the existence of augmented Lagrange multipliers for the constrained minimum problem (P): minimizef(x), subject tog(x)=0, is investigated under very general assumptions onX,f, andg. In the first part, we use the existence of augmented Lagrange multipliers as a sufficient condition for the convergence of the algorithm. In the second part, we prove that this is also a necessary condition for the convergence of the method and the boundedness of the sequence of the multiplier estimates. Further, we give very simple examples to show that the existence of augmented Lagrange multipliers is independent of smoothness condition onf andg. Finally, an application to the linear-convex problem is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933164
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  • 4
    Electronic Resource
    Electronic Resource
    A feasible conjugate-direction method to solve linearly constrained minimization problems (1975)
    Springer
    Journal of optimization theory and applications 16 (1975), S. 25-38 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; quadratically convergent algorithms ; conjugate-direction methods ; linearly constrained nonlinear programming ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An iterative procedure is presented which uses conjugate directions to minimize a nonlinear function subject to linear inequality constraints. The method (i) converges to a stationary point assuming only first-order differentiability, (ii) has ann-q step superlinear or quadratic rate of convergence with stronger assumptions (n is the number of variables,q is the number of constraints which are binding at the optimum), (iii) requires the computation of only the objective function and its first derivatives, and (iv) is experimentally competitive with well-known methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935621
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  • 5
    Electronic Resource
    Electronic Resource
    A method of multipliers for mathematical programming problems with equality and inequality constraints (1975)
    Springer
    Journal of optimization theory and applications 17 (1975), S. 155-161 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; method of multipliers ; penalty function methods ; inequality constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0. The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λ T φ(x)+kφ T (x)φ(x) −μ T $$\tilde \psi $$ (x)+k $$\tilde \psi $$ T (x) $$\tilde \psi $$ (x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and $$\tilde \psi $$ (x), and the elements of $$\tilde \psi $$ (x) are defined by $$\tilde \psi $$ (j)(x)=min[ψ(j)(x), (1/2k) μ(j)]. The scalark〉0 is the penalty constant, held fixed throughout the algorithm. Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933920
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  • 6
    Electronic Resource
    Electronic Resource
    On programming with absolute-value functions (1975)
    Springer
    Journal of optimization theory and applications 17 (1975), S. 181-183 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; absolute-value problems ; optimality criteria ; linear programming ; nonconvex programming ; operations research
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers some programming problems with absolute-value (objective) functions subject to linear constraints. Necessary and sufficient conditions for the existence of finite optimum solutions to these problems are proved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933924
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  • 7
    Electronic Resource
    Electronic Resource
    Converse duality in Banach space (1975)
    Springer
    Journal of optimization theory and applications 17 (1975), S. 229-238 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; converse duality ; Banach spaces ; convexity ; optimal control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two theorems on converse duality are obtained for mathematical programs in Banach spaces. The proofs are based on a Banach-space generalization of the F. John necessary condition for a constrained minimum. No use is made of Kuhn-Tucker constraint qualifications. In the second theorem, the primal program contains a nonlinear equality constraint, and a converse duality theorem is obtained, using a modified concept of convexity; this result appears new, even for finite-dimensional programs. The results are applied to a problem in optimal control.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933877
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  • 8
    Electronic Resource
    Electronic Resource
    Rates of convergence for a method of centers algorithm (1976)
    Springer
    Journal of optimization theory and applications 18 (1976), S. 199-228 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; penalty-function methods ; convergence rate ; method of centers
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Convergence of a method of centers algorithm for solving nonlinear programming problems is considered. The algorithm is defined so that the subproblems that must be solved during its execution may be solved by finite-step procedures. Conditions are given under which the algorithm generates sequences of feasible points and constraint multiplier vectors that have accumulation points satisfying the Fritz John or the Kuhn-Tucker optimality conditions. Under stronger assumptions, linear convergence rates are established for the sequences of objective function, constraint function, feasible point, and multiplier values.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935704
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  • 9
    Electronic Resource
    Electronic Resource
    On the Pironneau-Polak method of centers (1976)
    Springer
    Journal of optimization theory and applications 20 (1976), S. 269-295 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; inequality constraints ; numerical methods ; descent methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The method of centers is a well-known method for solving nonlinear programming problems having inequality constraints. Pironneau and Polak have recently presented a new version of this method. In the new method, the direction of search is obtained, at each iteration, by solving a convex quadratic programming problem. This direction finding subprocedure is essentially insensitive to the dimension of the space on which the problem is defined. Moreover, the method of Pironneau and Polak is known to converge linearly for finite-dimensional convex programs for which the objective function has a positive-definite Hessian near the solution (and for which the functions involved are twice continuously differentiable). In the present paper, the method and a completely implementable version of it are shown to converge linearly for a very general class of finite-dimensional problems; the class is determined by a second-order sufficiency condition and includes both convex and nonconvex problems. The arguments employed here are based on the indirect sufficiency method of Hestenes. Furthermore, the arguments can be modified to prove linear convergence for a certain class of infinite-dimensional convex problems, thus providing an answer to a conjecture made by Pironneau and Polak.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933624
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  • 10
    Electronic Resource
    Electronic Resource
    Extensions of Lemke's algorithm for the linear complementarity problem (1976)
    Springer
    Journal of optimization theory and applications 20 (1976), S. 397-416 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; complementarity problems ; computing methods ; linear systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Lemke's algorithm for the linear complementarity problem fails when a desired pivot is not blocked. A projective transformation overcomes this difficulty. The transformation is performed computationally by adjoining a new row to a schema of the problem and pivoting on the element in this row and the unit constant column. Two new algorithms result; some conditions for their success are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933128
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  • 11
    Electronic Resource
    Electronic Resource
    On the bottleneck assignment problem (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 451-458 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; minimax problems ; operations research ; optimization theorems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a new approach for solving the bottleneck assignment problem is presented. The problem is treated as a special class of permutation problems which we call max-min permutation problems. By defining a suitable neighborhood system in the space of permutations and designating certain permutations as critical solutions, it is shown that any critical solution yields a global optimum. This theorem is then used as a basis to develop a general method to solve max-min permutation problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933089
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  • 12
    Electronic Resource
    Electronic Resource
    Convergence proof of minimization algorithms for nonconvex functions (1977)
    Springer
    Journal of optimization theory and applications 23 (1977), S. 203-210 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; optimization theory ; minimization algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we prove a theorem of convergence to a point for descent minimization methods. When the objective function is differentiable, the convergence point is a stationary point. The theorem, however, is applicable also to nondifferentiable functions. This theorem is then applied to prove convergence of some nongradient algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933047
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  • 13
    Electronic Resource
    Electronic Resource
    Production and marketing decisions with multiple objectives in a competitive environment (1978)
    Springer
    Journal of optimization theory and applications 24 (1978), S. 449-458 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; Fritz John problem ; infinitely constrained optimization problems ; production and marketing decisions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents an application of the mathematical program which has been called the Fritz John problem to a situation in applied economics. A model is presented which enables a firm to maximize profit, subject to constraints on maintenance of market share in the face of collusive action by competitors and subject to resource constraints. An example problem is formulated and solved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932888
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  • 14
    Electronic Resource
    Electronic Resource
    Mathematical programming techniques in majorization (1978)
    Springer
    Journal of optimization theory and applications 25 (1978), S. 361-373 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; majorization ; goal programming ; assignment problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we are concerned with mathematical programs which construct nonmajorizing vectors for several specific majorization applications. In particular, we develop a linear integer program and two integer goal programs which solve the assignment majorization problem. We also develop a quadratic program for solving majorization problems which arise in probability and statistics. In the appendix, we present a general goal-programming algorithm for these, as well as others, goal programs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932899
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  • 15
    Electronic Resource
    Electronic Resource
    A remark on a regularity assumption for the mathematical programming problem in Banach spaces (1978)
    Springer
    Journal of optimization theory and applications 25 (1978), S. 375-381 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; Slater regularity condition ; Lagrange multipliers ; functional analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with a recently proposed Slater-like regularity condition for the mathematical programming problem in infinite-dimensional vector spaces (Ref. 1). The attractive feature of this constraint qualification is the fact that it can be considered as a condition only on theactive part of the constraint. We prove that the studied regularity condition is equivalent to the regularity assumption normally used in the study of the mathematical programming problem in infinite-dimensional vector spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932900
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  • 16
    Electronic Resource
    Electronic Resource
    A direct method for the general solution of a system of linear equations (1975)
    Springer
    Journal of optimization theory and applications 16 (1975), S. 429-445 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; conjugate-gradient methods ; variable-metric methods ; linear equations ; numerical methods ; computing methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A computationally stable method for the general solution of a system of linear equations is given. The system isA Tx−B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA T and the augmented matrix [A T,B] are of the same rankm, wherem≤n, so that the system is consistent and solvable. Whenm〈n, the method yields the minimum modulus solutionx m and a symmetricn ×n matrixH m of rankn−m, so thatx=x m+H my satisfies the system for ally, ann-vector. Whenm=n, the matrixH m reduces to zero andx m becomes the unique solution of the system. The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933852
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  • 17
    Electronic Resource
    Electronic Resource
    Infinitely constrained optimization problems (1976)
    Springer
    Journal of optimization theory and applications 19 (1976), S. 261-281 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; cutting-plane methods ; minimax problems ; nonconvex programming ; zero-sum games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A generalized cutting-plane algorithm designed to solve problems of the form min{f(x) :x ∈X andg(x,y) ∈ 0 for ally ∈Y} is described. Convergence is established in the general case (f,g continuous,X andY compact). Constraint dropping is allowed in a special case [f,g(·,y) convex functions,X a convex set]. Applications are made to a variety of max-min problems. Computational considerations are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00934096
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  • 18
    Electronic Resource
    Electronic Resource
    Rates of convergence of a one-dimensional search based on interpolating polynomials (1979)
    Springer
    Journal of optimization theory and applications 27 (1979), S. 187-203 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; line search procedures ; interpolating polynomials ; convergence rates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this study, we derive the order of convergence of line search techniques based on fitting polynomials, using function values as well as information on the smoothness of the function. Specifically, it is shown that, if the interpolating polynomial is based on the values of the function and its firsts−1 derivatives atn+1 approximating points, the rate of convergence is equal to the unique positive rootr n+1 of the polynomial $$D_{n + 1} (z) = z^{n + 1} - (s - 1)z^n - s\sum\limits_{j = 1}^n {z^{n - j} } .$$ For alln, r n is bounded betweens ands+1, which in turn implies that the rate can be increased by as much as one wishes, provided sufficient information on the smoothness is incorporated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933226
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