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  • nonlinear programming  (101)
  • linear programming  (45)
  • 1
    ISSN: 1573-2878
    Keywords: Linear systems ; linear programming ; convexity ; polyhedra ; invariance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem of confining the trajectory of a linear discrete-time system in a given polyhedral domain is addressed through the concept of (A, B)-invariance. First, an explicit characterization of (A, B)-invariance of convex polyhedra is proposed. Such characterization amounts to necessary and sufficient conditions in the form of linear matrix relations and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and does not require the computation of vertices. Such advantages are felt particularly in the computation of the supremal (A, B)-invariant set included in a given polyhedron, for which a numerical method is proposed. The problem of computing a control law which forces the system trajectories to evolve inside an (A, B)-invariant polyhedron is treated as well. Finally, the (A, B)-invariance relations are generalized to persistently disturbed systems.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1573-2878
    Keywords: nonlinear programming ; path-following methods ; differential equations ; projective algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new method for linearly constrained nonlinear programming is proposed. This method follows affine scaling paths defined by systems of ordinary differential equations and it is fully parallelizable. The convergence of the method is proved for a nondegenerate problem with pseudoconvex objective function. In practice, the algorithm works also under more general assumptions on the objective function. Numerical results obtained with this computational method on several test problems are shown.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Cauchy method ; steepest descent ; convex programming ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We discuss two issues related to the Cauchy algorithm. First, we use anArmijo search with constant α≥0.5 and show that the sequence isFejer convergent to the optimal set, and hence convergent. Second, we useexact line searches and show an example in which the sequence fails toconverge.
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  • 4
    ISSN: 1573-2878
    Keywords: Duality theory ; linear systems ; nonlinear programming ; convex programming ; distributed control systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of singular control problems involving amplitude constraints on the controls is examined. IfL ∞ is the space of control functionsU, the control constraint setS can be identified with the unit ball inL ∞. Now, for anyn ∈ (1, ∞), an analogous problem may be set up withL n forU and the unit ball inL n forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn → ∞ is investigated.
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  • 5
    ISSN: 1573-2878
    Keywords: Engineering design ; nonlinear programming ; convex programming ; optimization theorems ; approximation of functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A possible mathematical formulation of the practical problem of computer-aided design of electrical circuits (for example) and systems and engineering designs in general, subject to tolerances onk independent parameters, is proposed. An automated scheme is suggested, starting from arbitrary initial acceptable or unacceptable designs and culminating in designs which, under reasonable restrictions, are acceptable in the worst-case sense. It is proved, in particular, that, if the region of points in the parameter space for which designs are both feasible and acceptable satisfies a certain condition (less restrictive than convexity), then no more than 2 k points, the vertices of the tolerance region, need to be considered during optimization.
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  • 6
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; penalty function methods ; gradient projection methods ; convergence analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new programming algorithm for nonlinear constrained optimization problems is proposed. The method is based on the penalty function approach and thereby circumyents the necessity to maintain feasibility at each iteration, but it also behaves much like the gradient projection method. Although only first-order information is used, the algorithm converges asymptotically at a rate which is independent of the magnitude of the penalty term; hence, unlike the simple gradient method, the asymptotic rate of the proposed method is not affected by the ill-conditioning associated with the introduction of the penalty term. It is shown that the asymptotic rate of convergence of the proposed method is identical with that of the gradient projection method.
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  • 7
    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.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    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.
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  • 9
    ISSN: 1573-2878
    Keywords: Optimality conditions ; geometric programming ; nonlinear programming ; ordinary programming ; Kuhn-Tucker conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Generalizations of the Kuhn-Tucker optimality conditions are given, as are the fundamental theorems having to do with their necessity and sufficiency.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Geometric programming ; duality ; subsidiary problems ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this paper is not to derive new results, but rather to provide insight that will hopefully aid researchers involved in the design and coding of algorithms for geometric programs. The main contributions made here are: (i) a computationally useful interpretation of the Lagrange multipliers associated with the dual orthogonality constraints, (ii) a computationally useful interpretation of the Lagrange multiplier associated with the dual normality constraint, and (iii) an analysis of the much-avoided issue of subsidiary problems.
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