An introduction to nonlinear programming—I: Necessary and sufficient conditions

Parameter optimization problems arise in a wide variety of applications. The importance of the general problem has led to the investigation and development of the field of nonlinear programming and of the special classes of these problems such as linear programming. The study of these problems has reached a sophisticated level and an extensive literature exists, including a number of excellent textbooks. In this and in subsequent articles, the basic results of nonlinear programming are reviewed, developed, and discussed. No attempt is made here to survey the vast literature on the subject but sufficient references are included to provide the reader with a basis for a more detailed study. It is hoped that this series of tutorial articles will make the literature more accessible and will provide a useful perspective on the current status of the field. In this first article, the conditions which a solution must satisfy are discussed. To a lesser extent, conditions are examined which, when satisfied, ensure that a solution has been obtained. The discussion in this article of necessary and sufficient conditions for an optimum provides a basis for the subsequent examination of algorithms which can be used to determine the solution of parameter optimization (i.e. nonlinear programming) problems.     

Necessary and sufficient conditions for stability of LMS

Guo and Ljung (1995) established some general results on exponential stability of random linear equations, which can be applied directly to the performance analysis of a wide class of adaptive algorithms, including the basic LMS ones, without requiring stationarity, independency, and boundedness assumptions of the system signals. The current paper attempts to give a complete characterization of the exponential stability of the LMS algorithms by providing a necessary and sufficient condition for such a stability in the case of possibly unbounded, nonstationary, and non-φ-mixing signals. The results of this paper can be applied to a very large class of signals, including those generated from, e.g., a Gaussian process via a time-varying linear filter. As an application, several novel and extended results on convergence and the tracking performance of LMS are derived under various assumptions. Neither stationarity nor Markov-chain assumptions are necessarily required in the paper     

Necessary and sufficient conditions for existence of decouplingcontrollers

It is well known that if a linear time-invariant plant is free from coincidences of poles and zeros in the right half-plane, then it can be decoupled with internal stability under unity-feedback configuration. We consider plants for which such coincidences do occur and give necessary and sufficient conditions under which stabilizing decoupling controllers exist. The conditions derived, based on transfer matrices and residues, are simple and straightforward     

Necessary and sufficient conditions for consistent global snapshots

Consistent global snapshots are important in many distributed applications. We prove the exact conditions for an arbitrary checkpoint, or a set of checkpoints, to belong to a consistent global snapshot, a previously open problem. To describe the conditions, we introduce a generalization of Lamport's (1978) happened-before relation called a zigzag path     

Necessary and sufficient conditions for balancing unstable systems

Necessary and sufficient conditions are given for the existence of balancing transformations for minimal state-space realizations(A, B, C)whereAmay be unstable. These conditions are expressed in terms of the real diagonalizability of the product of the reachability Gramian and the observability Gramian. For symmetric realizations these conditions can be reformulated in terms of the real diagonalizability of the cross Gramian, and we show that minimal symmetric systems can be internally balanced if the associated Hankel matrix is positive semidefinite. Examples are given of minimal systems, including symmetric systems, which cannot be balanced.     

Necessary and sufficient conditions for local second-order identifiability

This note presents a new formulation and proof of the result the Hessian of the likelihood function of an observed process at the pointthetain a parameter space, computed under the assumption that the process is i.i.d. Gaussian, is asymptotically nonsingular if and only ifthetais locally second-order identifiable. That is to say, if and only if the parameters in a neighborhood ofthetaare in one-to-one correspondence with the second-order statistics of the observed process.     

Necessary and sufficient conditions for broadcast consensus protocols

Summary We consider consensus protocols in asynchronous distributed systems that are based on broadcast communication. We show that a necessary and sufficient condition for the existence of a deterministic consensus protocol is delivery of each broadcast message to at least (n+k+1)/2 processes in ann-process system subject tok crash failures with either eventual fair broadcasting or eventual full broadcasting. The broadcast model captures the idea of a broadcast communication medium, such as the Ethernet, in which messages, if delivered, are delivered immediately and in order but not necessarily to all processes. Louise E. Moser received the Ph.D. degree in mathematics from the University of Wisconsin, Madison, in 1970. From 1970 to 1987 she was a professor of mathematics and computer science at California State University, Hayward. In 1987 she moved to the University of California, Santa Barbara, where is currently on the faculty of the Department of Electrical and Computer Engineering. Her current research interests include parallel and distributed systems, network architectures and communication protocols, and formal methods in software engineering. P.M. Melliar-Smith received the Ph.D. degree in computer science from the University of Cambridge, Cambridge, England, in 1987. He was a senior research scientist and program director at SRI International in Menlo Park (1976–1987), senior research associate at the University of Newcastle Upon Tyne (1973–1976), and principal designer for GEC Computers Ltd. in England (1964–1973). He is currently a professor in the Department of Electrical and Computer Engineering at the University of California, Santa Barbara. His research interests include fault-tolerant distributed systems, high-speed communication networks and protocols, and formal specification and verification. Vivek Agrawala received the B.Tech. degree in chemical engineering in 1984 and the M. Tech. degree in computer technology in 1986, both from the Indian Institute of Technology, Delhi, and a Ph.D. in computer science in 1991 from the University of California, Santa Barbara. Since then he has been a Research Scientist at Siemens Corporate Research, Princeton, New Jersey. His major research interests are distributed algorithms, software design methods, and distributed systems.This research was supported by the National Science Foundation, Grant Numbers CCR-8908515 and NCR-9016361. V. Agrawala's current address is Siemens Corporate Research, 755 College Road East, Princeton, NJ 08540, USA     

Necessary and sufficient conditions for robust oscillatory stability

The problem of robust oscillatory stability of uncertain systems is investigated in this article. For the uncertain systems, whose characteristic polynomial sets belong to the interval polynomial family or diamond polynomial family, sufficient and necessary conditions are given based on the stability and/or oscillation properties of some special extreme point polynomials. A systematic approach exploiting Yang's complete discrmination system is proposed to check the robust oscillatory stability of such uncertain systems. The proposed method is efficient in computation and can be easily implemented.     

Necessary and sufficient conditions for decoupling using output feedback

Necessary and sufficient conditions for decoupling linear multivariable systems using output feedback are determined. For a system which satisfies these conditions, the class of all feedback matrices which decouple the system is characterized. These conditions provide realistic criteria for determining if a compensator or an observer is required for decoupling.     

Necessary and sufficient conditions for reachability on a simplex

In this paper we solve the general problem of designing a feedback controller to reach a set of facets of an n-dimensional simplex in finite time, for a system evolving with linear affine dynamics. Necessary and sufficient conditions are presented in the form of bilinear inequalities on the vertices of the simplex. By exploiting the structure of the problem, the bilinear inequalities are converted to a series of linear programming problems.     

Necessary and sufficient conditions for the stability of linear parameter-dependent systems

In this paper we discuss the necessary and sufficient conditions for the Hurwitz stability of a linear system whose matrix depends continuously on several parameters. We also give an algorithm based on our derived condition to determine the stability of the system in a finite number of steps. Numerical examples are given to illustrate the results.     

Necessary and sufficient conditions for strong comparability of multicomponent systems

In this article we analyze the stochastic comparison of continuous time Markov chains with state space E?=?{0,1,2,...} ^N used as models for N??component systems. Although the conditions for comparability of such processes are known in the theory, they involve a huge family of sets which makes their checking a difficult task. We show how, under some assumptions on the transitions allowed in the chains, the conditions can be greatly simplified, writing them in terms of conditions for each component. As an application we give the necessary and sufficient conditions for the stochastic comparability of two batch-arrival assemble-transfer queueing networks in terms of conditions for their single stations, improving previous results on the subject (Economou, J Appl Probab 40:1103–1120, 2003b). We also analyze a finite site version of interacting particle systems studied in Borrello (Electron J Probab 16:106–151, 2011).     

Necessary and sufficient graphical conditions for formation control of unicycles

The feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control. The directed graph defined by the information flow plays a key role. It is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node. A similar result is given for formation stabilization to a line and to more general geometric arrangements.     

Necessary and sufficient conditions for the Hurwitz and Schurstability of interval matrices

Establishes a set of new sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors use these results to establish necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors relate the above results to the existence of quadratic Lyapunov functions for linear time-invariant systems with interval-valued coefficient matrices. Using the above results, the authors develop an algorithm to determine the Hurwitz and the Schur stability properties of interval matrices. The authors demonstrate the applicability of their results by means of two specific examples     

Necessary and sufficient optimality conditions for discrete time-invariant automaton-type systems

The optimal control of deterministic discrete time-invariant automaton-type systems is considered. Changes in the system’s state are governed by a recurrence equation. The switching times and their order are not specified in advance. They are found by optimizing a functional that takes into account the cost of each switching. This problem is a generalization of the classical optimal control problem for discrete time-invariant systems. It is proved that, in the time-invariant case, switchings of the optimal trajectory (may be multiple instantaneous switchings) are possible only at the initial and (or) terminal points in time. This fact is used in the derivation of equations for finding the value (Hamilton–Jacobi–Bellman) function and its generators. The necessary and sufficient optimality conditions are proved. It is shown that the generators of the value function in linear–quadratic problems are quadratic, and the value function itself is piecewise quadratic. Algorithms for the synthesis of the optimal closed-loop control are developed. The application of the optimality conditions is demonstrated by examples.     

Necessary and sufficient convergence conditions of the instrumental variable method for identification

The consistency of the instrumental variable method is considered for a system driven by independent identically distributed input signals with zero mean. Necessary and sufficient conditions for convergence are developed with respect to the given model structure. The sufficient and necessary conditions are established for cases with and without data prefiltering. In this paper we extend on existing results such that the requirements on the degree of the instrumental variables can be relaxed. Numerical examples supporting the theoretical results are provided.