Proof Of Optimality Research Articles

Managing search complexity in linguistic geometry

This paper is a new step in the development of linguistic geometry. This formal theory is intended to discover and generalize the inner properties of human expert heuristics, which have been successful in a certain class of complex control systems, and apply them to different systems. In this paper, we investigate heuristics extracted in the form of hierarchical networks of planning paths of autonomous agents. Employing linguistic geometry tools the dynamic hierarchy of networks is represented as a hierarchy of formal attribute languages. The main ideas of this methodology are shown in the paper on two pilot examples of the solution of complex optimization problems. The first example is a problem of strategic planning for the air combat, in which concurrent actions of four vehicles are simulated as serial interleaving moves. The second example is a problem of strategic planning for the space comb of eight autonomous vehicles (with interleaving moves) that requires generation of the search tree of the depth 25 with the branching factor 30. This is beyond the capabilities of modern and conceivable future computers (employing conventional approaches). In both examples the linguistic geometry tools showed deep and highly selective searches in comparison with conventional search algorithms. For the first example a sketch of the proof of optimality of the solution is considered.

A short proof of optimality of the Bottom Up algorithm for discrete resource allocation problems

The paper deals with the problem of maximizing a separable concave function over integer points of a polymatroid, known to be the integer case of the resource allocation problem. The Bottom Up algorithm for the case when the polymatroid is specified by an explicit list of constraints was presented by H. Groenevelt (European Journal of Operational Research 54 (1991) 227–236). We give a new proof of optimality of the Bottom Up algorithm. It is considerably shorter and simpler than the original proof. Our prood is based on the analysis of the greedy algorithm for this problem and properties of greedy solutions.

State Feedback Minimax and Guaranteed Cost Controllers for Linear Systems with Integral Quadratic Constraint Uncertainty Descriptions

This paper considers the design of minimax and optimal guaranteed cost controllers for uncertain linear systems. The uncertainty is permitted to have a structured description and is bounded by an integral quadratic constraint. For known initial conditions on the system state, it is shown that minimax control can be realised by static full state feedback controllers. A new proof of minimax optimality is given using standard methods for linear systems. The design of such controllers requires the optimal solution of a single Riccati equation dependent upon a set of parameters whose dimension is equal to the number of uncertainty blocks. It is shown that this multivariable optimisation is convex and thus numerically tractable. No explicit criteria to determine the existence of a solution are known; however, if solutions exist, results pertaining to their location are given to assist numerical optimisation. The extension of the approach to arbitrary unknown initial conditions is considered but is shown, in general, not to be possible. A tractable approach, considering the initial condition to be a random variable whose occurrence satisfies a certain probability density function, is presented. The resulting optimisation problem is shown to be convex and thus numerically tractable.

Optimality of the double exponential formula - functional analysis approach -

In the light of the functional analysis theory we establish the optimality of the double exponential formula. The argument consists of the following three ingredients: (1) introduction of a number of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity; (2) proof of the (near-) optimality of the trapezoidal formula in each space introduced in (1) by showing the (near-) equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures; (3) nonexistence theorem for the spaces, the characterizing decay rate of which is more rapid than the double exponential.

Some connections between isoperimetric and Sobolev-type inequalities

Introduction Differential and integral forms of isoperimetric inequalities Proof of Theorem 1.1 A relation between the distribution of a function and its derivative A variational problem The discrete version of Theorem 5.1 Proof of propositions 1.3 and 1.5 A special case of Theorem 1.2 The uniform distribution on the sphere Existence of optimal Orlicz spaces Proof of Theorem 1.9 (the case of the sphere) Proof of Theorem 1.9 (the Gaussian case) The isoperimetric problem on the real line Isoperimetry and Sobolev-type inequalities on the real line Extensions of Sobolev-type inequalities to product measures on $\mathbf{R}^{n}$ References.

Adaptive decentralized control under non-uniqueness of the optimal control

We study the problem of decentralization of flow control in packet-switching networks under the isarithmic scheme. An incoming packet enters the network only if there are permits available at the entry port when it arrives. The actions of the controllers refer to the routing of permits in the network and the control variables are the corresponding probabilities. We study the behavior of adaptive algorithms implemented at the controllers to update these probabilities and seek optimal performance. This problem can be stated as a routing problem in a closed queueing network. The centralized version of a learning automation is a general framework presented along with the proof of asymptotic optimality. Decentralization of the controller gives rise to non-uniqueness of the optimal control parameters. Non-uniqueness can be exploited to construct asymptotically optimal learning algorithms that exhibit different behavior. We implement two different algorithms for the parallel operation and discuss their differences. Convergence is established using the weak convergence methodology. In addition to our theoretical results, we illustrate the main results using the flow control problem as a model example and verify the predicted behavior of the two proposed algorithms through computer simulations, including an example of tracking.

On the Approximation of Continuous Functions by Fourier–Legendre Sums

Let {Xn}∞0be the orthonormal system of Legendre polynomials on [−1, 1]. Forf∈C[−1, 1] letSn(f) (n+1∈N) be thenth partial sum of the Fourier–Legendre series of the functionf. Some refinements of the classical inequality[formula]involving best approximation inLp-norms are discussed. For a class of examples we obtain better order estimates than those that can be derived from (1). Furthermore, we show that the results are best possible in a certain sense. It turns out that only in two particular cases (p=43andp=4) there is no proof of optimality of the results. In conclusion, we give without proof a generalization of the main theorem to the ultraspherical case.

Optimal Nonlinear Estimation of Linear Stochastic Systems: The Multivariable Extension

This paper extends to multi-input multi-output (MIMO) systems a nonlinear method of simultaneous parameter and state estimation that appeared in the ASME JDSM&C (September, 1994), for single-input single-output (SISO) systems. The method is called pseudo-linear identification (PLID), and applies to stochastic linear time-invariant discrete-time systems. No assumptions are required about pole or zero locations; nor about relative degree, except that the system transfer functions must be strictly proper. In the earlier paper, proofs of optimality and convergence were given. Extensions of those proofs to the MIMO case are also given here.

Supporting stored video

VBR compressed video is known to exhibit significant, multiple-time-scale bit rate variability. In this paper, we consider the transmission of stored video from a server to a client across a high speed network, and explore how the client buffer space can be used most effectively toward reducing the variability of the transmitted bit rate.We present two basic results. First, we present an optimal smoothing algorithm for achieving the greatest possible reduction in rate variability when transmitting stored video to a client with given buffer size. We provide a formal proof of optimality, and demonstrate the performance of the algorithm on a set of long MPEG-1 encoded video traces. Second, we evaluate the impact of optimal smoothing on the network resources needed for video transport, under two network service models: Deterministic Guaranteed service [1, 9] and Renegotiated CBR (RCBR) service [8, 7]. Under both models, we find the impact of optimal smoothing to be dramatic.

On the Composition of Zero-Knowledge Proof Systems

The wide applicability of zero-knowledge interactive proofs comes from the possibility of using these proofs as subroutines in cryptographic protocols. A basic question concerning this use is whether the (sequential and/or parallel) composition of zero-knowledge protocols is zero-knowledge too. We demonstrate the limitations of the composition of zero-knowledge protocols by proving that the original definition of zero-knowledge is not closed under sequential composition; and that even the strong formulations of zero-knowledge (e.g., black-box simulation) are not closed under parallel execution. We present lower bounds on the round complexity of zero-knowledge proofs, with significant implications for the parallelization of zero-knowledge protocols. We prove that three-round interactive proofs and constant-round Arthur-Merlin proofs that are black-box simulation zero-knowledge exist only for languages in BPP. In particular, it follows that the “parallel versions” of the first interactive proofs systems presented for quadratic residuosity, graph isomorphism, and any language in NP, are not black-box simulation zero-knowledge, unless the corresponding languages are in BPP Whether these parallel versions constitute zero-knowledge proofs was an intriguing open questions arising from the early works on zero-knowledge. Other consequences are a proof of optimality for the round complexity of various known zero-knowledge protocols and the necessity of using secret coins in the design of “parallelizable” constant-round zero-knowledge proofs.

Asymptotically Minimax Nonparametric Regression in L2

Additive nonparametric regression with equidistant observation design is considered. The Pinsker-type minimax results are derived and the linear asymptotically minimax estimators are exhibited, based on approximation of the initial nonparametric model by a linear models of dimension which is increasing with the number of observations. The proof of optimality of these linear estimators within the class of all possible estimators is based on the rather elementary but very useful van Trees inequality.

Scheduling a two-machine robotic cell: A solvable case

The paper deals with the scheduling of a robotic cell in which jobs are processed on two tandem machines. The job transportation between the machines is done by a transportation robot. The robotic cell has limitations on the intermediate space between the machines for storing the work-in-process. What complicates the scheduling problem is that the loading/unloading operation times are non-negligible. Given the total number of operationsn, an optimalO(n logn)-time algorithm is proposed together with the proof of optimality.

A correction and an addendum for "Variable-rate coding for meteor-burst communications"

A correction is presented in this note for an error which appears in the subject paper by M.B. Pursley and S.D. Sandberg [ibid., vol. 37, pp. 1105-1 112, Nov. 1989]. In addition, attached as an addendum is a proof that a modified version of an algorithm presented in the subject paper produces an optimal packet con6guration for variable-rate coding in meteor-burst communications. The proof of optimality for the original algorithm is quite tedious, and was not included in the subject paper.

Optimal Proofs of Determinacy

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L(ℝ).The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL(ℝ) ⇒ HODL(ℝ) ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals.

An optimality proof for asynchronous recovery algorithms in distributed systems

We prove the optimality of asynchronous recovery algorithms for distributed systems in the sense that irrespective of the order of roll backs by sites, all sites incrementally converge to a unique consistent cut which is the “latest” of all past consistent cuts formed by the local recovery points of the sites.

Uniform modal damping of rings by an extended node control theorem

Hazelrigg, G. A., The Use of Green's Theorem to Find Global Optimal Solutions to a Class of Impulsive Transfer, American Astronautical Society, Paper 68-092, Sept. 1968. Marec, J. P., Optimal Space Trajectories, Elsevier, Amsterdam, 1979. Palmore, J. I., An Elementary Proof of the Optimality of Hohmann Transfers, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 5, 1984. Prussing, J. E., Simple Proof of the Global Optimality of the Hohmann Transfer, Journal of Guidance, Control, and Dynamics, Vol. 15, No. 4, 1992.

Optimal incremental parsing

This communication sets the problem of incremental parsing in the context of a complete incremental compiling system. It turns out that, according to the incrementally paradigm of the attribute evaluator and data-flow analyzer to be used, two definitions of optimal incrementality in a parser are possible. Algorithms for achieving both forms of optimality are given, both of them based on ordinary LALR(1) parse tables. Optimality and correctness proofs, which are merely outlined in this communication, are made intuitive thanks to the concept of awell-formed list of threaded trees, a natural extension of the concept ofthreaded treefound in earlier works on incremental parsing.

Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

Universal bounds for the cardinality of codes in the Hamming space F/sub r//sup n/ with a given minimum distance d and/or dual distance d' are stated. A self-contained proof of optimality of these bounds in the framework of the linear programming method is given. The necessary and sufficient conditions for attainability of the bounds are found. The parameters of codes satisfying these conditions are presented in a table. A new upper bound for the minimum distance of self-dual codes and a new lower bound for the crosscorrelation of half-linear codes are obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Completeness and optimality of marginal likelihood estimating equations

A counter-example shows that the proof of optimality of the marginal likelihood estimating function for parameter of interest, under the conditions assumed in Lloyd (1987), contains a gap and is, thus, invalid. The same comment applies to the generalized version of Lloyd’s Theorem given by Bhapkar and Srinivasan (1993). In the light of known results concerning Fisher information for parameter of interest and partial sufficiency of a suitable statistic, the counter-example reveals a similar gap in the proof of corollary 3.2 of Bhapkar (1991).

Improved optimal algorithms for scheduling unit-length independent tasks on uniform machines

Five scheduling problems are considered, concerning unit-length independent tasks and uniform machines. New improved optimal algorithms are presented that can solve these problems in at most O( n log n) time, where n is the number of tasks. The existing algorithms solve most of these problems in O( n 3 ) time. Proofs of optimality of the present algorithms are included, and simple representative examples are provided that illustrate the type of results obtained