Set Of Multipliers Research Articles

Comparison on wavelets techniques for solving fractional optimal control problems

This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton's method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods.

Critical solutions of nonlinear equations: stability issues

It is known that when the set of Lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton, this set may contain so-called critical multipliers. This special subset of Lagrange multipliers defines, to a great extent, stability pattern of the solution in question subject to parametric perturbations. Criticality of a Lagrange multiplier can be equivalently characterized by the absence of the local Lipschitzian error bound in terms of the natural residual of the optimality system. In this work, taking the view of criticality as that associated to the error bound, we extend the concept to general nonlinear equations (not necessarily with primal–dual optimality structure). Among other things, we show that while singular noncritical solutions of nonlinear equations can be expected to be stable only subject to some poor “asymptotically thin” classes of perturbations, critical solutions can be stable under rich classes of perturbations. This fact is quite remarkable, considering that in the case of nonisolated solutions, critical solutions usually form a thin subset within all the solutions. We also note that the results for general equations lead to some new insights into the properties of critical Lagrange multipliers (i.e., solutions of equations with primal–dual structure).

An efficient computational method based on the hat functions for solving fractional optimal control problems

In this paper, an efficient and accurate computational method based on the hat functions (HFs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the fractional optimal control problem under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional state and control variables are expanded by the HFs with unknown coefficients. Then, the operational matrix of fractional integration of the HFs with some properties of these basis functions are employed to achieve a nonlinear algebraic equation, replacing the performance index and a nonlinear system of algebraic equations, replacing the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

Wavelets method for solving fractional optimal control problems

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the FOCP under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional derivative of the state variable and the control variable are expanded by the LWs with unknown coefficients. Then, the operational matrix of the Riemann–Liouville fractional integration with some properties of the LWs are employed to achieve a nonlinear algebraic equation, in place of the performance index and a linear system of algebraic equations, in place of the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extrema, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers is applied. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

Behaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurations

Augmented Lagrangian methods represent an efficient way to carry out the forward-dynamics simulation of mechanical systems. These algorithms introduce the constraint forces in the dynamic equations of the system through a set of multipliers. While most of these formalisms were obtained using Lagrange’s equations as starting point, a number of them have been derived from Hamilton’s canonical equations. Besides being efficient, they are generally considered to be robust, which makes them especially suitable for the simulation of systems with discontinuities and impacts. In this work, we have focused on the simulation of mechanical assemblies that undergo singular configurations. First, some sources of numerical difficulties in the proximity of singular configurations were identified and discussed. Afterwards, several augmented Lagrangian and Hamiltonian formulations were compared in terms of their robustness during the forward-dynamics simulation of two benchmark problems. Newton–Raphson iterative schemes were developed for these formulations with the Newmark formula as numerical integrator. These outperformed fixed point iteration approaches in terms of robustness and efficiency. The effect of the formulation parameters on simulation performance was also assessed.

Necessary Optimality Conditions for Optimal Control Problems with Equilibrium Constraints

This paper introduces and studies the optimal control problem with equilibrium constraints (OCPEC). The OCPEC is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint, and it can be seen as a dynamic mathematical program with equilibrium constraints. It provides a powerful modeling paradigm for many practical problems such as bilevel optimal control problems and dynamic principal-agent problems. In this paper, we propose weak, Clarke, Mordukhovich, and strong stationarities for the OCPEC. Moreover, we give some sufficient conditions to ensure that the local minimizers of the OCPEC are Fritz John--type weakly stationary, Mordukhovich stationary, and strongly stationary. Unlike Pontryagain's maximum principle for the classical optimal control problem with equality and inequality constraints, a counterexample shows that for general OCPECs there may exist two sets of multipliers for complementarity constraints. A condition under which these...

Second Order Analysis for Strong Solutions in the Optimal Control of Parabolic Equations

In this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations. We consider first the case of box constraints on the control in a general setting and then, in the case of a quadratic Hamiltonian, we also impose final integral constraints on the state. For the first problem, in a rather general framework, we prove a characterization of the quadratic growth property in the strong sense, i.e., for admissible controls whose associated states are uniformly near to the state of the nominal control. Assuming a quadratic Hamiltonian in the state constrained case, we provide a sufficient second order optimality condition for the aforementioned quadratic growth property, which does not impose the uniqueness of the Lagrange multiplier set. As a consequence of our results, in the quadratic Hamiltonian case, under some continuity assumptions on the data we prove that the notions of quadratic growth in the strong sense coincide with the more standard notion of quadratic growth in the weak sense, i.e., with respect to controls which are uniformly near to the nominal one.

Truncated Predictor Feedback for Periodic Linear Systems With Input Delays With Applications to the Elliptical Spacecraft Rendezvous

This paper considers truncated predictor feedback (TPF) stabilization of periodic linear systems with time-varying input delays with its applications to the elliptical spacecraft rendezvous. By carefully investigating the solutions and properties to a class of parametric Lyapunov differential equations, it is proved that TPF controllers can stabilize the periodic linear time-delay systems with arbitrarily large and bounded delays as long as the characteristic multiplier set of the open-loop system is within the closed unit circle. Both single input delay and multiple input delays are addressed. Moreover, the control matrices of the systems are not necessarily periodic. The developed TPF approach is then applied to the elliptical spacecraft rendezvous problem, and numerical simulations validate the effectiveness of the established approaches.

Boundedness of certain sets of Lagrange multipliers in vector optimization

In this paper, we establish Lagrange multiplier rules in terms of Michel–Penot subdifferential for nonsmooth vector optimization problem. A constraint qualification or regularity condition in terms of Michel–Penot subdifferential is given and under this regularity condition the boundedness of certain sets of Lagrange multipliers are discussed.

Economic impacts of the hotel industry: an input-output analysis

Purpose– The purpose of this paper is to be limited to provide an overall economic structure linked with a particular hospitality industry by identifying the economic structure of relations between the two hotel industries and other industries within a particular state in the USA.Design/methodology/approach– The analyses of output, labor income and employment multipliers from the input–output system were performed using the IMPLAN 3.0 software. The study attempted to compare the hotel/motel industry (industry code 411) and the accommodations industry (industry code 412) with the top ten industries and averages of each set of multipliers to estimate the relative importance and contribution of the two hotel industries to the economy of Texas. After this comparison, the aggregated input–output tables and multipliers were prepared to determine the economic inter-relationship between the two combined hotel industries (industry code 411 plus industry code 412) and the non-hotel industries, using the criteria of the NAICS (North American Industry Classification System).Findings– The three findings of this study are summarized as follows. First, the two hotel industries impacted the state economy due to a high induced effect from output and a considerable direct, indirect and induced effect from labor income and employment, despite their relatively lower multipliers and the economic downturn in the state. Second, the hotel-related industry had a strong inter-dependent relationship with the finance and insurance-related industries. Finally, while the hotel industry generated more labor income and employment than did the other accommodations industry, it is interesting that the other accommodations industry created more output than did the hotel industry.Research limitations/implications– Other than limitations pertaining to assumptions of input-output model, an input-output analysis alone cannot become the best analytical method for decision-making. The study was a cross-sectional study with 2009 data and did not incorporate a time-series flow of the state economic structure over several decades. A study of the inter-relationship among varied states bordering the state could be worthwhile to identify the flow of inputs and outputs.Originality/value– Despite a considerable number of research in measuring the economic impacts, this paper was of great significance, in that the economic impact of the hotel industry that has never been performed in a particular state of the USA was analyzed. Additionally, these quantified economic data and results should be helpful to future plans and policies associated with the hotel industry.

Robust H 2 and H ∞ filtering for discrete‐time uncertain Linear fractional transform systems

This study is concerned with the problem of robust H 2 and H ∞ filtering for uncertain discrete-time linear systems. Two types of time-invariant parametric uncertainty, namely polytopic and ellipsoidal, are considered and represented by a linear fractional transformation structure. Obtained auxiliary variables by a convex optimisation problem, play the role of decoupling the Lyapunov variables and the robust filter parameters, in order to cast the problem into a linear matrix inequality-based optimisation problem. The design conditions are derived based on the quadratic separation concept and employing appropriate parameterisations for the corresponding set of multipliers. The merit of the methods presented in this study lies in their less conservatism than the existing methods for the polytopic uncertain systems, as well as presenting a new convex optimisation procedure for the robust filtering for the ellipsoidal uncertain systems.

Transient Simulation of Multiscale Structures Using the Nonconformal Domain Decomposition Laguerre-FDTD Method

In this paper, a full-wave transient domain decomposition method is proposed for solving multiscale electromagnetic problems. The proposed scheme is based on the unconditionally stable Laguerre finite-difference time-domain (Laguerre-FDTD) method. The entire computational domain is partitioned into several subdomains with independent meshing according to the physical sizes of each subdomain. Standard Laguerre-FDTD method is used to form the interior field equations of each subdomain, whereas interface field equations are formed by applying the equivalency between finite-element method and FDTD method. In addition, field continuity is enforced through the use of two sets of Lagrange multipliers at the domain interface. Schur complement is implemented for extracting the interface problem and each domain can be evaluated independently. Numerical results for multiscale structures are presented to demonstrate the capability, accuracy, and efficiency of the proposed method.

A dual domain decomposition method for finite element digital image correlation

SummaryThe computational burden associated to finite element based digital image correlation methods is mostly due to the inversion of finite element systems and to image interpolations. A non‐overlapping dual domain decomposition method is here proposed to rationalise the computational cost of high resolution finite element digital image correlation measurements when dealing with large images. It consists in splitting the global mesh into submeshes and the reference and deformed states images into subset images. Classic finite element digital image correlation formulations are first written in each subdomain independently. The displacement continuity at the interfaces is enforced by introducing a set of Lagrange multipliers. The problem is then condensed on the interface and solved by a conjugate gradient algorithm. Three different preconditioners are proposed to accelerate its convergence. The proposed domain decomposition method is here exemplified with real high resolution images. It is shown to combine the metrological performances of finite element based digital image correlation and the parallelisation ability of subset based methods. Copyright © 2015 John Wiley & Sons, Ltd.

Comments on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it

We discuss a certain special subset of Lagrange multipliers, called critical, which usually exist when multipliers associated to a given solution are not unique. This kind of multipliers appear to be important for a number of reasons, some understood better, some (currently) not fully. What is clear, is that Newton and Newton-related methods have an amazingly strong tendency to generate sequences with dual components converging to critical multipliers. This is quite striking because, typically, the set of critical multipliers is “thin” (the set of noncritical ones is relatively open and dense, meaning that its closure is the whole set). Apart from mathematical curiosity to understand the phenomenon for something as classical as the Newton method, the attraction to critical multipliers is relevant computationally. This is because convergence to such multipliers is the reason for slow convergence of the Newton method in degenerate cases, as convergence to noncritical limits (if it were to happen) would have given the superlinear rate. Moreover, the attraction phenomenon shows up not only for the basic Newton method, but also for other related techniques (for example, quasi-Newton, and the linearly constrained augmented Lagrangian method). Despite clear computational evidence, proving that convergence to a critical limit must occur appears to be a challenge, at least for general problems. We outline the partial results obtained up to now. We also discuss the important role that noncritical multipliers play for stability, sensitivity, and error bounds. Finally, an important issue is dual stabilization, i.e., techniques to avoid moving along the multiplier set towards a critical one (since it leads to slow convergence). We discuss the algorithms that do the job locally, i.e., when initialized close enough to a noncritical multiplier, their dual behavior is as desired. These include the stabilized sequential quadratic programming method and the augmented Lagrangian algorithm. However, when the starting point is far, even those algorithms do not appear to provide fully satisfactory remedies. We discuss the challenges with constructing good algorithms for the degenerate case, which have to incorporate dual stabilization for fast local convergence, at an acceptable computational cost, and also be globally efficient.

When are minimizing controls also minimizing relaxed controls?

Relaxation refers to the procedure of enlarging the domain of a variational problem or the search space for the solution of a set of equations, to guarantee the existence of solutions. In optimal control theory relaxation involves replacing the set of permissible velocities in the dynamic constraint by its convex hull. Usually the infimum cost is the same for the original optimal control problem and its relaxation. But it is possible that the relaxed infimum cost is strictly less than the infimum cost. It is important to identify such situations, because then we can no longer study the infimum cost by solving the relaxed problem and evaluating the cost of the relaxed minimizer. Following on from earlier work by Warga, we explore the relation between the existence of an infimum gap and abnormality of necessary conditions (i.e. they are valid with the cost multiplier set to zero). Two kinds of theorems are proved. One asserts that a local minimizer, which is not also a relaxed minimizer, satisfies an abnormal form of the Pontryagin Maximum Principle. The other asserts that a local relaxed minimizer that is not also a minimizer satisfies an abnormal form of the relaxed Pontryagin Maximum Principle.

Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem

When there is uncertainty in the lower level optimization problem of a bilevel programming, it can be formulated by a robust optimization method as a bilevel program with lower level second-order cone programming problem (SOCBLP). In this paper, we show that the Lagrange multiplier set mapping of the lower level problem of a class of the SOCBLPs is upper semicontinuous under suitable assumptions. Based on this fact, we detect the similarities and relationships between the SOCBLP and its KKT reformulation. Then we derive the specific expression of the critical cone at a feasible point, and show that the second order sufficient conditions are sufficient for the second order growth at an M-stationary point of the SOCBLP under suitable conditions.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L2(G)) of bounded linear operators on L2(G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E⊆G is a set of multiplicity if and only if the set E⁎={(s,t)∈G×G:ts−1∈E} is a set of operator multiplicity. Analogous results are established for M1-sets and M0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function ψ:G→C defines a closable multiplier on the reduced C⁎-algebra Cr⁎(G) of G if and only if Schur multiplication by the function N(ψ):G×G→C, given by N(ψ)(s,t)=ψ(ts−1), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L2(G). Similar results are obtained for multipliers on VN(G).

A Lagrange duality approach to state-feedback stabilisability in switched positive linear systems

Lagrange duality is used in order to obtain a novel and useful state-feedback exponential stabilisability characterisation for discrete-time switched positive linear systems. A switched positive linear system is proved to be state-feedback exponentially stabilisable if and only if there is a set of Lagrange multipliers whose image, via a linear mapping of the system's modes, is a Schur matrix. This characterisation directly leads to a simple and novel method for the synthesis of exponentially stabilising state-feedback mappings. A particular set of Lagrange multipliers is explicitly obtained as the result of constructively establishing an upper bound on the minimum number of Lagrange multipliers required to attain (through the aforementioned linear mapping) a Schur matrix. This construction of a particular set of Lagrange multipliers can always be used to complete the first step of the obtained synthesis method. It is further shown that for a (stabilisable) switched positive linear system with rank-one modes, the number of modes that constitute the system is an upper bound on the minimum number of Lagrange multipliers required to generate a Schur matrix, as well as an upper bound on the minimum number of linear functionals required to represent an exponentially stabilising state-feedback mapping. Examples illustrating the obtained method for the synthesis of stabilising state-feedback mappings are included.

Assessing the impact of fiscal measures on the Czech economy

Abstract We build a DSGE model to investigate the transmission of fiscal policy to the real economy in the Czech Republic. Departing from the elements of the Czech National Bank's current g3 forecasting model (Andrle, Hledik, Kamenik, and Vlcek, 2009), we introduce a comprehensive fiscal sector incorporating a number of modeling features that are often neglected in the mainstream DSGE literature, e.g. allowing government consumption and government capital to be productive. Furthermore, we extend our fiscal model to include unemployment in a way proposed by Gali, Lopez-Salido, and Valles (2007). Crucial fiscal parameters, related mainly to the specified fiscal rule, are estimated using Bayesian techniques. The model is then used to calculate a set of fiscal multipliers for individual revenue and expenditure items of the government budget. We find that the largest real GDP fiscal multipliers in the first year are associated with government consumption (0.6), social security contributions paid by employers (0.6), and government investment (0.5).

EFFECTS OF SECTOR AGGREGATION ON CO2 MULTIPLIERS IN MULTIREGIONAL INPUT–OUTPUT ANALYSES

The past few years have seen the emergence of several global multiregional input–output (MRIO) databases. Due to the cost and complexity of developing such extensive tables, industry sectors are generally represented at a rather aggregate level. Currently, one of the most important applications of input–output analysis is environmental assessments, for which highly aggregate sectors may not be sufficient to yield accurate results. We experiment with four of the most important global MRIO systems available, analyzing the sensitivity of a set of aggregate CO2 multipliers to aggregations in the MRIO tables used to calculate them. Across databases, we find (a) significant sensitivity to background system detail and (b) that sub-sectors contained within the same aggregate MRIO sector may exhibit highly different carbon multipliers. We conclude that the additional information provided by the extra sector detail may warrant the additional costs of compilation, due to the heterogeneous nature of economic sectors in terms of their environmental characteristics.