Necessary Conditions For Minimum Research Articles

Necessary conditions for the extremum in non-smooth problems of variational calculus

In the paper, we proposed an approach for studying strong and weak extremums in non-smooth vector problems of calculus of variation, namely, in classic variational problems with fixed ends and with a free right end, and also in a variational problem with higher derivatives. The essence of the proposed approach is to introduce a Weierstrass type variation characterized by a numerical parameter. Necessary conditions for minimum containing as corollaries the Weierstrass condition, its local modification and also the Legendre and transversality conditions are obtained. In the case when the Legendre condition degenerates, equality and inequality type necessary conditions are obtained for the weak local minimum. The examples showing the content-richness of the obtained main results are given.

Optimality Conditions in Terms of Alternance: Two Approaches

In Optimization Theory, necessary and sufficient optimality conditions play an essential role. They allow, first of all, checking whether a point under study satisfies the conditions, and, secondly, if it does not, finding a "better" point. For the class of directionally differentiable functions, a necessary condition for an unconstrained minimum requires the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form of inclusion. The problem of verifying this condition is reduced to that of finding the point of some convex and compact set C which is nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction, and are able to construct a numerical method. In the classical Chebyshev polynomial approximation problem, necessary optimality conditions are expressed in the form of alternation of signs of some values. In the present paper, a generalization of the alternance approach to a general optimization problem is given. Two equivalent forms of the alternance condition (the so-called inside form and the outside one) are discussed in detail. In some cases, it may be more convenient to use the conditions in the form of inclusion, in some other--the condition in the alternance form as in the Chebyshev approximation problem. Numerical methods based on the condition in the form of inclusion usually are "gradient-type" methods, while methods employing the alternance form are often "Newton-type". It is hoped that in some cases it will be possible to enrich the existing armory of optimization algorithms by a new family of efficient tools. In the paper, we discuss only unconstrained optimization problems in the finite-dimensional setting. In many cases, a constrained optimization problem can be reduced (via Exact Penalization Techniques) to an unconstrained one.

Minimality of Variable-Thrust Subarcs in Optimal Chemical Rocket Trajectories

F ROM the beginning of rocket trajectory optimization, it has been known that the optimal trajectory of a rocket can be composed of three types of subarcs: maximum thrust, variable thrust, and zero thrust. To determine whether or not a particular subarc can be part of aminimum-fuel trajectory, it is possible to apply some tests (necessary conditions). Necessary conditions for a relative minimum are classified according to the types of the controls, that is, regular controls (nonsingular), singular controls, andmixed controls (regular and singular), and according to level, that is, weak or strong. Weak conditions are the Legendre–Clebsch condition [1] for regular controls, the generalized Legendre–Clebsch condition [2] for singular controls, andGoh’s condition [3] for mixed controls. Strong conditions are the Weierstrass condition [1] for regular controls, the equivalent of the Weierstrass conditon [4] for singular controls, and the equivalent of Goh’s condition for mixed controls (yet to be derived). A subarc is defined to be “minimizing” if the relevant necessary conditions for a minimum are satisfied. Although this use of the term minimizing might seem to be unconventional to some researchers, it is not. See for example Bliss [5] (p. 20), Bryson and Ho [6] (p. 247), Kopp and Moyer [7] (p. 1443), Goh [3] (p. 721), and Bell and Jacobson [2] (p. 3). For a chemical rocket, the particular variable-thrust subarc, known as Lawden’s spiral, was claimed by Lawden to be minimizing by applying the Weierstrass condition [8] (p. 112). Subsequently, Robbins [9], Kelley [10] , Kopp andMoyer [7], and Kelley et al. [11] developed the generalized Legendre–Clebsch condition and applied it to theLawden spiral. Their conclusionwas that theLawden spiral is not minimizing. The same conclusion was reached by Kelley [12] using state transformations. Later, Lawden [13] wrote that the variable-thrust subarc is not minimizing but did not say why the Weierstrass condition gave a conflicting result. This Note has two purposes. The first is to review the preceding necessary conditions and their use. The second is to explain why Lawden’s application of the Weierstrass condition did not eliminate the variable-thrust subarc.

Optimal Industrial Controller Tuning Algorithms in View of Constraints for Stability Margins

This paper presents a technique for optimal H 2 tuning of a fixed-order and fixed-structure controller for SISO plant represented by a state-space realization. The proposed technique is based on minimization of some H 2 proximity criterion for transfer functions of closed-loop control system and its implicit reference model (internal model control) subject to restrictions onto H ∞ norm of transfer function of closed-loop system guaranteeing its stability. Proposed tuning algorithm for the controller parameters uses estimates of the plant parameters obtained via parametric identification. It is shown that necessary conditions for minimum of H 2 norms of open and closed-loop systems coincide with necessary conditions for minimum of Frobenius norm of a matrix tuning polynomial linearly depending on the controller parameters. The proposed technique is illustrated by a numerical example.

Nonsmooth optimization of hydrothermal problems

In this paper the authors present a necessary condition for minimum of a functional J ( z ) ≔ ∫ 0 T L ( t , z ( t ) , z ′ ( t ) ) d t in the case in which the function L is continuous but not of class C 1 . This situation arises in problems of optimization of hydrothermal systems with pumped-storage plants. In such problems, the function L z ′ ( t , z , · ) is discontinuous in z ′ = 0 , which is the borderline point between the power generation zone ( z ′ > 0 ) and the pumping zone ( z ′ < 0 ) . The problem can be naturally formulated in the framework of nonsmooth analysis, using the generalized (or Clarke's) gradient.

One remark to Ekeland's variational principle

This article deals with the generalization of Ekeland's first-order necessary conditions of minimum. They are extended to include second-order conditions. The results are applied to variational calculus and mathematical physics.

Optimal Planning of a Collision-free Trajectory of Redundant Manipulators

In this study, global planning of a collision-free trajectory (simultaneous planning of geometric path and its time para meterization) involving potential functions and direct kinematic and dynamic equations of the manipulator is presented. This planning is based on using the necessary conditions of min imum for an integral type criterion. General transversality conditions corresponding to the boundary ones (resulting from the task to be performed) are derived. Thus, closed systems of boundary dependences fully specifying differential equations, which arise from the necessary conditions of extreme for the above criterion, are obtained. As a result, this system renders it possible to reduce the collision-free trajectory planning problem to a two-point boundary value problem. A numerical example involving a manipulator of three-revolute kinematic pairs of the Vth class is considered.

Uniqueness conditions for Kuhn-Tucker points on a disk

Consider minimizingf onD which is diffeomorphic to a disk. Under a genericity assumption, the number of points onD satisfying the Kuhn-Tucker necessary conditions for minimum is odd. We give conditions which imply that a local minimum is global and a necessary and sufficient condition that a Kuhn-Tucker point is the solution. Convex transformable problems satisfy the latter condition.D may be of full dimension or be embedded on a manifold or it may be given by a system of concave inequalities.

Second Order Necessary Conditions in Optimization

It is known that if a restricted minimization problem satisfies first order necessary conditions for minimum at some point with multiple choices of Lagrange multiplier vectors (or linear functionals) then, in general, second order conditions for different critical variations may require different Lagrange multipliers. We present here a relatively simple derivation of new second order necessary conditions in which different critical variations share a common Lagrange multiplier if they are “pairwise critical”. The problems that we consider contain restrictions in the form of finitely many equalities and of (possibly infinite-dimensional) inclusions involving arbitrary convex bodies. These new conditions generalize in some respects previous results of Dennis S. Bernstein (A systematic approach to higher-order necessary conditions in optimization theory, SIAM J. Control Optim., 22 (1984), pp. 211–238).

Controllability, extremality, and abnormality in nonsmooth optimal control

We derive sufficient conditions for controllability and necessary conditions for minimum in nonsmooth optimal control problems defined by differential or functional-integral equations with isoperimetric and unilateral restrictions. We consider the cases when the controls are relaxed or chosen fromabundant sets of original (ordinary) controls (which include most, or all, of the control sets studied in the literature). We prove that, if there exist optimal strictly original controls (that is, controls that are optimal in an abundant set but not among relaxed controls), then the problem admits abnormal extremals. We also study the abnormality of the optimal strictly original controls themselves.

Estimacion bayesiana multiple de un parametro

The problem to be analyzed in this paper deals with the finding ofn valuesx 1,x 2, …,x n∈IR which minimizes the function\(E[\mathop { \min c}\limits_{ i = 1 ... n} (\xi - x_i )]\) where ξ is a one-dimensional random variable with known distribution function ϕ andc is a measurable and positive function.First, conditions onc in order to ensure the existence of a solution to this problem are determined. Next, necessary conditions to be satisfied by the point (x 1,x 2, …,x n) in which the function attains the minimum are looked for. Finally, conditions on ϕ are given to guarantee the uniqueness of the point which verifies the necessary conditions for minimum, when the functionc isc(y)=|y| andc(y)=y 2.

Necessary conditions for minimum in problems with nonstandard cost functionals

Usual formulation of optimal control problems involves the minimization of a cost functional which is of the form of a definite integral. In this chapter we develop necessary conditions for an optimal control in the case of problems in which the cost functional is either a quotient or a product of definite integrals. We call such functionals nonstandard, and these naturally arise in Chapters 2,4, and 5 during the computation of the performance of a suboptimal H∞ controller. In Chapters 2,4, and 5 a criterion for the evaluation of the cost functional will be presented in the specialized case of linear systems and quadratic integrands. Preliminary results for problems having a fixed final time and free terminal state are in [1]. Related results can also be found in [2,3]. In this monograph, we consider only fixed final time problems. Problems in which the final time is free are treated in [4]. In Section 5, we discuss the relation of our results to those in [2,3].

Restricted Minima of Functions of Controls

Previous article Restricted Minima of Functions of ControlsJ. WargaJ. Wargahttps://doi.org/10.1137/0305043PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. Warga, Functions of relaxed controls, SIAM J. Control, 5 (1967), 628–641 10.1137/0305042 MR0226474 0234.49007 LinkGoogle Scholar[2] J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111–128 10.1016/0022-247X(62)90033-1 MR0142020 0102.31801 CrossrefGoogle Scholar[3] J. Warga, Necessary conditions for minimum in relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 129–145 10.1016/0022-247X(62)90034-3 MR0142021 0102.31802 CrossrefGoogle Scholar[4] Lucien W. Neustadt, An abstract variational theory with applications to a broad class of optimization problems. I. General theory, SIAM J. Control, 4 (1966), 505–527 10.1137/0304040 MR0216349 0166.09401 LinkGoogle Scholar[5] Lucien W. Neustadt, An abstract variational theory with applications to a broad class of optimization problems. II. Applications, SIAM J. Control, 5 (1967), 90–137 10.1137/0305007 MR0237210 0172.12903 LinkGoogle Scholar[6] E. J. McShane, Necessary conditions in generalized-curve problems of the calculus of variations, Duke Math. J., 7 (1940), 1–27 10.1215/S0012-7094-40-00701-3 MR0003478 0024.32503 CrossrefGoogle Scholar[7] Hubert Halkin, On the necessary condition for optimal control of non-linear systems, J. Analyse Math., 12 (1964), 1–82 MR0171653 0128.10103 CrossrefGoogle Scholar[8] S. Saks, Theory of the Integral, Monografje Matematyczne, New York, 1937, reprinted by Hafner Publishing Co. 0017.30004 Google Scholar[9] N. Dunford and , J. Schwarz, Linear Operators I, Interscience, New York, 1964 Google Scholar Previous article FiguresRelatedReferencesCited byDetails Jack Warga: In Appreciation14 July 2006 | SIAM Journal on Control and Optimization, Vol. 33, No. 2AbstractPDF (809 KB) Volume 5, Issue 4| 1967SIAM Journal on Control History Submitted:22 May 1967Accepted:28 August 1967Published online:18 July 2006 InformationCopyright © 1967 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0305043Article page range:pp. 642-656ISSN (print):0036-1402Publisher:Society for Industrial and Applied Mathematics

Optimal stratification according to the method of least squares

Abstract A one-dimensional random variable X is given. We have L points, µ1, µ2, …, µ L , and define the random variable Z = minµ h | X — µ h |, that is the distance to the nearest of the L points µ1, …, µ L . We want to find that set of points µ h for which the function has a minimum. As we shall see in section 2, this problem is equivalent to finding L strata with the set of points of stratification x 1, x 2, …, x L−1 that makes a minimum. wh is the probability mass and σ2 h the variance of the hth stratum. By differentiation of φ with respect to xh one can show [3] that a necessary condition for minimum is where µh is the mean of the hth stratum. In section 2 we obtain this condition in another way, which at the same time gives a method of finding the points µh and xh .

Necessary conditions for minimum in relaxed variational problems

Necessary conditions for minimum in relaxed variational problems

Variational problems of optimization of control processes for equations with discontinuous right-hand sides

In a variational formulation, the problens of optimization of control processes are discussed for systems described by differential equations with discontinuous right-hand sides. The necessary conditions of minimum of a functional are established and then applied to solution of the problems of optimization of the states of motion of oscillating conveyors.

On variational problems of optimization of control processes

The problems of optimization of control processes in recent years attract considerable attention of researchers. The classical apparatus of the calculus of variations [ 1–5 ] as well as newer methods are employed for their solution. Certain results, important for the theory of optimum systems, have been obtained with the use of the maximum principle of Pontriagin [ 6–9 ], the methods of functional analysis [ 10 ], and the method of dynamic programming [ 11 ]. Numerous questions of optimization of control processes can be formulated in the form of the Lagrange problem [ 1, 12–15 ], the Mayer problem [ 6,8 ], and the Mayer-Bolza problem of the calculus of variation. Here, the most general of them, the Mayer-Bolza problem, is discussed, with the modification introduced by the questions of optimization [ 2–4, 13, 14 ] and with the limitations imposed on the controls being taken into account. For this case the necessary conditions of minimum are established. The author expresses his gratitude to A.I. Lur'e for his attention and help in carrying out this work.