A chance-constrained programming algorithm

This paper describes a new algorithm solving the deterministic equivalents of chance-constrained problems where the random variables are normally distributed and independent of each other. In this method nonlinear chance-constraints are first replaced by uniformly tighter linear constraints. The resulting linear programming problem is solved by a standard simplex method. The linear programming problem is then revised using the solution data and solved again until the stopping rule of the algorithm terminates the process. It is proved that the algorithm converges and that the solution found is the -optimal solution of the chance-constrained programming problem.The computational experience of the algorithm is reported. The algorithm is efficient if the random variables are distributed independently of each other and if they number less than two hundred. The computing system is called CHAPS, i.e. Chance-ConstrainedProgrammingSystem.     

A value iteration method for undiscounted multichain Markov decision processes

This paper proposes a value iteration method which finds an-optimal policy of an undiscounted multichain Markov decision process in a finite number of iterations. The undiscounted multichain Markov decision process is reduced to an aggregated Markov decision process, which utilizes maximal gains of undiscounted Markov decision sub-processes and is formulated as an optimal stopping problem. As a preliminary, sufficient conditions are presented under which a policy is-optimal.

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Zusammenfassung In dieser Arbeit wird eine Wertiterationsmethode vorgeschlagen, die eine-optimale Politik für einen undiskontierten nicht-irreduziblen Markovschen Entscheidungsprozeß (MEP) in endlichen vielen Schritten liefert. Der undiskontierte nicht-irreduzible MEP wird auf einen aggregierten MEP reduziert, der maximale Gewinn eines undiskontierten Sub-MEP verwendet und als optimales Stopp-Problem formuliert wird. Zu Beginn werden hinreichende Bedingungen für die-Optimalität einer Politik angegeben.

     

On controlling the parameter in the logarithmic barrier term for convex programming problems

We present a log-barrier based algorithm for linearly constrained convex differentiable programming problems in nonnegative variables, but where the objective function may not be differentiable at points having a zero coordinate. We use an approximate centering condition as a basis for decreasing the positive parameter of the log-barrier term and show that the total number of iterations to achieve an -tolerance optimal solution isO(|log()|)×(number of inner-loop iterations). When applied to then-variable dual geometric programming problem, this bound becomesO(n ² U/), whereU is an upper bound on the maximum magnitude of the iterates generated during the computation.The authors gratefully acknowledge very constructive and insightful comments and suggestions from the two anonymous referees and the correspondence from A. V. Fiacco (Ref. 1).     

Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison

We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P: find a globally-optimal value off and a corresponding point; Problem Q: find a set of disjoint subintervals of [a, b] containing only points with a globally-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P. In phase I, this algorithm obtains rapidly a solution which is often globally-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves the-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globally-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For small, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.The research of the authors has been supported by AFOSR grants 0271 and 0066 to Rutgers University. Research of the second author has been also supported by NSERC grant GP0036426, FCAR grant 89EQ4144 and partially by AFOSR grant 0066. We thank Nicole Paradis for her help in drawing the figures.     

On the complexity of approximating a KKT point of quadratic programming

We present a potential reduction algorithm to approximate a Karush—Kuhn—Tucker (KKT) point of general quadratic programming (QP). We show that the algorithm is a fully polynomial-time approximation scheme, and its running-time dependency on accuracy (0, 1) is O((l/) log(l/) log(log(l/))), compared to the previously best-known result O((l/)²). Furthermore, the limit of the KKT point satisfies the second-order necessary optimality condition of being a local minimizer. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research support in part by NSF grants DDM-9207347 and DMI-9522507, and the Iowa Business School Summer Grant.     

Asymptotics of the first correction in the perturbation of theN-soliton solution to the KdV equation

We consider a triple Fourier-type integral that represents a solution to the KdV equation linearized on anN-soliton potential. Assuming that the parameters of the potential depend on the slow timet, we construct an asymptotics of this integral as 0 uniform with respect tox, t up to large timet ^–1.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 204–217, August, 1995.The work was financially supported in part by the Russian Foundation for Basic Research under grant No. 94-01-00193a.     

Potential reduction method for harmonically convex programming

In this paper, we introduce a potential reduction method for harmonically convex programming. We show that, if the objective function and them constraint functions are allk-harmonically convex in the feasible set, then the number of iterations needed to find an -optimal solution is bounded by a polynomial inm, k, and log(1/). The method requires either the optimal objective value of the problem or an upper bound of the harmonic constantk as a working parameter. Moreover, we discuss the relation between the harmonic convexity condition used in this paper and some other convexity and smoothness conditions used in the literature.The authors like to thank Dr. Hans Nieuwenhuis for carefully reading this paper and the anonymous referees for the worthy suggestions.     

Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the strong duality theorem and derive a dual-to-primal conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find an-optimal solution for the quadratically constrained minimum cross-entropy analysis.     

A convergence analysis for a convex version of Dikin's algorithm

This paper is concerned with the convergence property of Dikin's algorithm applied to linearly constrained smooth convex programs. We study a version of Dikin's algorithm in which a second-order approximation of the objective function is minimized at each iteration together with an affine transformation of the variables. We prove that the sequence generated by the algorithm globally converges to a limit point at a local linear rate if the objective function satisfies a Hessian similarity condition. The result is of a theoretical nature in the sense that in order to ensure that the limit point is an -optimal solution, one may have to restrict the steplength to the order ofO(). The analysis does not depend on non-degeneracy assumptions.     

On the classical logarithmic barrier function method for a class of smooth convex programming problems

In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an -optimal solution isO((1+M ²) log) orO((1+M ²)nlog), depending on the updating scheme for the lower bound.on leave from Eötvös University, Budapest, Hungary.     

The correctness problem for best approximations by trigonometric polynomials in the class W o r H[ω]C

Suppose thatk, rz₊, W _o ^r H[]_C= {ff is a 2-periodic function,f C^r [–, ], (f^(r), ) ()}, T_k is the space of trigonometric polynomials of order k, p_k(f)T_k is the polynomial of best uniform approximation to f, and E_k(f) is the error of the best approximation. It is shown that for an arbitrary > 0 we have,where for 0<&#x2A7D;(1),k > 0.R () is the root of the equation , and for k = 0 or > (1) we have R()=.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 85–101, July, 1977.The author thanks S. B. Stechkin for posing the problem and for his attention to this work.     

Controlled dual perturbations for l p -programming

l _p -programming is a common generalization of linear programming, quadratically constrained quadratic programming,l _p -constrainedl _p -approximation, and multiple criteria compromise programming. It is a type of convex programming with objective function and inequality constraints expressed by means ofl _p -norms. The dual program established by Peterson and Ecker is a maximization problem with a concave, upper-semicontinuous objective function over a set of constraints that are essentially linear. In developing a dual method for this problem, we face two major difficulties. One is the non-differentiability of the dual objective function and the other one is an efficient dual-to-primal conversion.In this paper, we introduce a mechanism to construct a suitably perturbed dual program with a differentiable concave objective function over linear constraints. Solving this well-constructed perturbed dual program, we can obtain an-optimal dual solution for an arbitrarily small number. Moreover, we show a way of constructing a linear program based on this dual solution. Then an-optimal primal solution can be obtained by solving the dual of this simple linear program.

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Zusammenfassung Diel _p -Optimierung ist eine Verallgemeinerung, die die lineare Optimierung, quadratische Optimierung mit quadratischen Restriktionen,l _p -Approximation mitl _p -Restriktionen, wie auch Vektoroptimierung umfaßt. Es handelt sich dabei um konvexe Optimierungsaufgaben, bei denen Zielfunktions- und Ungleichungsrestriktionen mittelsl _p -Normen ausgedrückt werden. Das duale Problem nach Peterson and Eckert ist ein Maximierungsproblem mit einer konkaven oberhalb-halbstetigen Zielfunktion über einer Menge von im wesentlichen linearen Restrictionen. Bei der Entwicklung einer dualen Lösungsmethode treten zwei Hauptschwierigkeiten auf: Die eine ist die Nicht-Differenzierbarkeit der dualen Zielfunktion, die andere besteht darin, eine effiziente Übertragung der dualen Lösung in eine primale zu finden.In dieser Arbeit führen wir eine Methode ein, die es gestattet, ein entsprechendes gestörtes duales Programm mit differenzierbarer konkaver Zielfunktion und linearen Restriktion aufzustellen. Bei der Lösung dieses wohl-strukturierten, gestörten dualen Problems erhalten wir eine-optimale Duallösung für beliebig kleines. Ferner zeigen wir einen Weg auf, wie, basierend auf dieser Duallösung, ein lineares Programm formuliert werden kann. Löst man das Dualproblem dieses einfachen linearen Programms, so erhält man eine-optimale Lösung für das Ausgangsproblem.

     

ε-optimality criteria for convex programming problems via exact penalty functions

In this paper, we present-optimality criteria for convex programming problems associated with exact penalty functions. Several authors have given various criteria under the assumption that such convex problems and the associated dual problems can be solved. We assume the solvability of neither the convex problem nor the dual problem. To derive our criteria, we estimate the size of the penalty parameter in terms of an-solution for the dual problem.     

An efficient algorithm for solving semi-infinite inequality problems with box constraints

Many design objectives may be formulated as semi-infinite constraints. Examples in control design, for example, include hard constraints on time and frequency responses and robustness constraints. A useful algorithm for solving such inequalities is the outer approximations algorithm. One version of an outer approximations algorithm for solving an infinite set of inequalities(x, y) 0 for allyY proceeds by solving, at iterationi of the master algorithm, a finite set of inequalities ((x, y) 0 for allyY _i) to yieldx _i and then updatingY _i toY _i+1=Y _i {y_i } wherey _i arg max {(x _i,y)¦y Y}. Since global optimization is computationally extremely expensive, it is desirable to reduce the number of such optimizations. We present, in this paper, a modified version of the outer approximations algorithm which achieves this objective.The research reported herein was sponsored by the National Science Foundation Grants ECS-9024944, ECS-8816168, the Air Force Office of Scientific Research Contract AFOSR-90-0068, and the NSERC of Canada under Grant OGPO-138352.     

Polynomiality of an inexact infeasible interior point algorithm for semidefinite programming

In this paper we present a primal-dual inexact infeasible interior-point algorithm for semidefinite programming problems (SDP). This algorithm allows the use of search directions that are calculated from the defining linear system with only moderate accuracy, and does not require feasibility to be maintained even if the initial iterate happened to be a feasible solution of the problem. Under a mild assumption on the inexactness, we show that the algorithm can find an -approximate solution of an SDP in O(n²ln(1/)) iterations. This bound of our algorithm is the same as that of the exact infeasible interior point algorithms proposed by Y. Zhang.Research supported in part by the Singapore-MIT alliance, and NUS Academic Research Grant R-146-000-032-112.Mathematics Subject Classification (1991): 90C05, 90C30, 65K05     

A Combined Feasible-Infeasible Point Continuation Method for Strongly Monotone Variational Inequality Problems

In this paper, we discuss the variational inequality problems VIP(X, F), where F is a strongly monotone function and the convex feasible set X is described by some inequaliy constraints. We present a continuation method for VIP(X, F), which solves a sequence of perturbed variational inequality problems PVIP(X, F, , ) depending on two parameters 0 and >0. It is worthy to point out that the method will be a feasible point type when =0 and an infeasible point type when >0, i.e., it is a combined feasible–infeasible point (CFIFP for short) method. We analyse the existence, uniqueness and continuity of the solution to PVIP(X, F, , ), and prove that any sequence generated by this method converges to the unique solution of VIP(X, F). Moreover, some numerical results of the algorithm are reported which show the algorithm is effective.     

Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.     

An Analytic Center Cutting Plane Method for Solving Semi-Infinite Variational Inequality Problems

We study a variational inequality problem VI(X,F) with X being defined by infinitely many inequality constraints and F being a pseudomonotone function. It is shown that such problem can be reduced to a problem of finding a feasible point in a convex set defined by infinitely many constraints. An analytic center based cutting plane algorithm is proposed for solving the reduced problem. Under proper assumptions, the proposed algorithm finds an -optimal solution in O*(n ²/²) iterations, where O*(·) represents the leading order, n is the dimension of X, is a user-specified tolerance, and is the radius of a ball contained in the -solution set of VI(X,F).     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

Numerische Behandlung singulärer Sturm-Liouville-Probleme

Summary Generalizing the method of Wendroff [9] and using an estimate for the square integral of a normed eigenfunction outside a compact set, bounds are obtained for the eigenvalues of singular Sturm-Liouville problems from a finite difference method. The number of mesh points necessary to obtain. the accuracy behaves like ^–&frac; ln if tends to zero. Some numerical examples are given.