Inexact Approach Research Articles

Equity-driven facility location: A two-stage robust optimization approach

This paper explores the computational challenge of incorporating equity in p-median facility location models under uncertain demand and discusses how two-stage robust programming can be employed to address the challenge. Our research evaluates various equity measures appropriate for facility location modeling and proposes a novel approach to reformulating the problem into a two-stage robust optimization framework, enhancing computational efficiency caused by incorporating equity and uncertainty into these models. We provide two solution algorithms: an exact and an inexact column-and-constraint generation (C&CG) method. Our findings suggest that although the exact C&CG method generally outperforms the inexact approach, both methods perform well when the number of variables is small, with the inexact C&CG demonstrating a slight advantage in computational time. We further conduct a detailed evaluation of the tractability of our reformulated model and the effectiveness of various equity measures through a real-world case study of Metro Vancouver.

A sustainable road pricing oriented bilevel optimization approach under multiple environmental uncertainties

The rapidly-increasing urban transportation contributes to a number of externalities such as congestion and pollution; moreover, environmental uncertainties widely exist and bring challenges to creating traffic emission control strategies. To address these problems, this paper proposes an inexact bilevel programming approach under stochastic and fuzzy uncertainties (BLP-SF) for a toll scheme design with the considerations of the externalities of vehicular emission and road pricing policy. The BLP-SF approach can deal with multiple environmental uncertainties by (1) specifying the uncertainties as probability distributions and/or fuzzy sets, and (2) integrating chance-constrained programming and fuzzy possibility programming into the bilevel programming framework. A road pricing problem is exemplified under various policy scenarios to demonstrate the applicability of the BLP-SF approach. It is shown that the proposed BLP-SF approach can achieve optimal charging schemes with improvements in total emission reduction and congestion alleviation compared with traditional models. This study contributes to urban transportation management associated with congestion and emission.

An Examination of P. Davidson’s Misunderstandings about the Limiting Frequency Theory of Probability, Ergodicity and Non Ergodicity: These Approaches Require that either the Number of Observations Approaches Infinity or Time Approaches Infinity in Order for the Limit to Exist

Paul Davidson’s technical understanding of the mathematical details of the Limiting Frequency theory of probability and Kolmogorov’s measure theoretic extension from additivity to countable additivity, which allows for an extension of the concept of the Law of Large Numbers to the concept or ergodicity and non ergodicity, where time is substituted for observations, is extremely poor. Davidson has repeatedly made errors in many journal articles published in the Journal of Post Keynesian Economics and books published by Edward Elgar about these concepts since 1979. For instance, one example of a fundamental error made by Davidson since 1982 is his assertion, contradicted by all mathematical statisticians, that “Non stationarity is a sufficient, but not a necessary, condition for nonergodicity”. Davidson has repeated this false claim many times in the literature since 1982 when it first appeared in an article published in the Journal of Post Keynesian Economics. It is very simple to grasp what Davidson’s fundamental error is from just the title of a 1973 paper published in The Annals of Probability by R W Madsen and D L Isaacson, titled “Strongly Ergodic Behavior for Non-Stationary Markov Processes”, to realize that the correct mathematical statement is that non-stationary processes can be ergodic. This means that the correct analysis, known for certain by all mathematical statisticians since 1973, is that “Non stationarity is a necessary, but not a sufficient, condition for nonergodicity”. There is no existing evidence that Paul Davidson had any knowledge of Keynes’s approach to measurement, which rejected the exact and precise approach of Kolmogorov and Tinbergen except as a very special case, in favor of an inexact approach to measurement based on approximation and imprecise, interval valued probability and outcomes. Given that the concepts of ergodicity and non ergodicity require as a necessary condition the acceptance of the measure theoretic approach of Kolmogorov,it is straightforward to conclude that Keynes would reject the concepts of ergodicity and non ergodicity in macroeconomics or in regard to the concept of uncertainty because Part II of the TP establishes non additivity and non linearity as the general case and additivity and linearity as a special case, which is what the Kolmogorov axioms require.

Killing ‘Homo Economicus’ (Jeremy Bentham) Is Much, Much, Much, More Easier Said Than Done: Only the Smith – Keynes Inexact Measurement Approach Can Neutralize the Strong Attraction of Bentham’s Mathematical Rational Utility Maximization Model

The tremendous attraction and allure of Bentham’s original rational, utility maximizing, calculator model, which Bentham wrote out in plain English so as to capture as large an audience as possible, to economists has been greatly underestimated by the opponents of the ‘Homo Economicus’ model, which was the creation of Jeremy Bentham in 1787, the same year that he launched his attacks on both Adam Smith’s The Theory of Moral Sentiments (1759) and The Wealth of Nations (1776). Benthamite Utilitarian ethics allows a practitioner to appear to be a hard scientist because he/she will be using a lot of equations and numbers, what can be called mathematical benefit – cost analysis, in his/her argument, as opposed to the user of duty ethics or virtue ethics, who will not be using any such mathematical approach. Bentham‘s shrewd, in depth understanding and grasp of the desire for approbation, recognition, and applause on the part of economists, who wanted to be held in the same regard as physicists, leads directly to the neoclassical economics of Max U, which is simply a mathematical translation of Bentham’s already completely worked out concept of a rational calculator, adding marginal utility and diminishing marginal utility to utility maximization. New neoclassical economists then added the rational expectations-real business cycle-Dynamic stochastic General Equilibrium approach, based on the Normal and Log Normal probability distributions, which extends Bentham’s basic static pendulum-oscillation, external, exogenous shock model to intertemporal analysis. Bentham recognized that Smith’s emphasis on an inexact, interval valued, imprecise approach to probability, due to the fact of uncertain evidence (1776,pp.105-113,228-245,419-423,714), had to be undermined if his utilitarian ethics approach, based on precise and exact utility maximization, was to succeed in replacing Smith’s Virtue ethics approach. Bentham similarly recognized that he had to attack and neutralize Smith’s emphasis on the internal, endogenous threat to the macro economy that came from a certain segment of the upper income class that Smith labelled as prodigals, imprudent risk takers and projectors, in order to replace Smith’s virtue ethics emphasis on the virtues of prudence and temperance, which directly conflicted with Bentham’s emphasis on utility maximization and claim that the love of money was an insatiable desire of all men. The virtues of prudence and temperance require the complete total rejection of any insatiability assumption. J. M Keynes, following Smith’s approach, put forth a nearly identical version of Smith’s original inexact approach to measurement that was based on Boole’s upper – lower probabilities approach. Keynes’s inexact approach to probability in Part II of the A Treatise on Probability, and statistics in part V, which he used in the General Theory as the foundation for his liquidity preference theory of the rate of Interest, which he called approximation in chapter 4 of the General Theory, directly challenged the neoclassical followers of Bentham (Jevons, Marshall, Pareto, Cournot, Walras, Frisch, Tinbergen), who relied on an exact, precise approach to probability and statistics. Killing ‘Homo Economicus’ (Jeremy Bentham) can only be accomplished by adopting the Smith- Keynes inexact approach to measurement in order to challenge the Classical-Neo Classical-New Neo Classical exact approach to measurement. Max U collapses if the probabilities are not precise and exact, so that they sum to one, where the whole is the linear, additive sum of all of the individual parts. Currently, there is no credible, intellectual threat from any heterodox school of economics to Bentham, a genius with an IQ estimated to be in the 180-200 range, who could be compared to Lord Sauron of Mordor in the Lord of the Rings Trilogy.

Keynes’s Philosophy of Mathematics Can’t Be Mastered by Reading the Notes of His Economics Lectures Made by His Students in the Years 1932–35

J M Keynes’s approach to the use of mathematics is based on his interactions with Bertrand Russell, Alfred North Whitehead, William Ernest Johnson, C. D. Broad, G. E. Moore, and Alfred Marshall. His views were carefully expressed in the A Treatise on Probability and General Theory. Keynes favored using systems of simultaneous equations both in the A Treatise on Probability and General Theory. A study of the student notes taken in the years from 1932 to 1935 alone will lead to the wrong answer regarding Keynes’s views on the role of mathematics and systems of equations in economics. Keynes’s philosophy can be expressed by the term Inexact Measurement or Approximation, which Keynes made the foundation for his inexact approach to measurement in probability, mathematics, and statistics.

J.M. Keynes’s Revolution in the General Theory in Macroeconomics is Based on a Foundation of His Inexact Measurement Approach in Parts II and V of the a Treatise on Probability

Keynes’s revolution in macroeconomic theory is based on his application in the General Theory of his Inexact measurement approach from the A Treatise on Probability. This involves the use of interval valued probability and interval valued outcomes. Keynes called this approach approximation. It entails nonadditive (sub and super) probabilities, imprecise probabilities, indefinite and indeterminate probabilities,to which Keynes added interval valued outcomes in the General Theory. Except in special cases, Keynes rejected both Exact measurement,which involved the use of precise, definite, determinate, additive, linear probabilities, which imply that the decision maker knows an exact probability distribution. Keynes developed Boole's original logical theory of probability by developing an alternative method of solving Boole’s problems. Like Boole, Keynes’s approach involved solving systems of equations, the constraints, using linear and non linear, mathematical programming techniques. Keynes uses special aggregation assumptions in order to use and present an exact mathematical analysis in his D-Z model of chapter 20, which is the foundation for his IS-LM(LP) model of chapter 21. The D-Z model deals with expectations and uncertainty that results in a specific Y value. That Y value is an actual value that is certain, quantitative, and deterministic. Keynes then combines Y with r, the nominal, long run rate of interest, to derive the IS and LM(LP) curves on pp.298-299 of the General Theory. The General Theory has, unfortunately, attracted two completely different types of interpreters, both of which conflict directly with Keynes’s use of inexact measurement. The first type of interpreter essentially follows the views of Hicks(1937), Tinbergen(1940), Modigliani(1944) and Klein(1947), for example, in asserting that Keynes's approach in the General Theory was, in fact, a purely verbal, literary, prose approach, with no mathematical or statistical foundation whatsoever, that needed to be reworked in order to present it in mathematical form. The second approach follows the Joan Robinson and G LS Shackle nihilist approach in asserting that Keynes argued that no mathematical, technical, analytic or quantitative approaches involving mathematical modeling or equations could ever be possible in macroeconomics because of the existence of what they called unknowledge or fundamental uncertainty, which meant that no decision maker has any knowledge whatsoever of anything that can occur in the future, be it the near future or the distant future, be it one day in the future, one month in the future, one year in the future, five years in the future, ten years in the future, twenty years in the future, etc. The best that could be done was to assume that decision makers could sometimes make use of a weak, ordinal probability approach. Both approaches have no support whatsoever in anything actually written by Keynes in his lifetime. The nihilistic approach of the Keynesian Fundamentalists (Moggridge, Skidelsky, Meeks, O’Donnell, Carabelli, Fitzgibbons, Runde, Davis, Winslow, etc.) is based on a gross misinterpretation of the diagram on page 39 of the A Treatise on Probability. Once this error is corrected, it is clear that Keynes used a mathematical approach involving non linear and non additive analysis in his approach to lower and upper probabilities that Tinbergen, Modigliani, and Klein simply overlooked because they were not familiar with the concept of interval valued probability. Keynes’s Inexact approach was evolved and can be studies in the work of Hailperin (1986) and Walley (1991).There are no macroeconomists who are familiar with this work. Keynes described his pp.298-299 model in a summary in the GT using the following words or phrases: • One can calculate a quantitative answer • The model is composed of three elements a), b),and c) • It provides an analysis that is valuable in introducing order and method to the enquiry • It is a set of simultaneous equations • It provides a determinate answer or result. • This result is an equilibrium Keynes is describing his IS-LM(LP) model in the General Theory.

The Role of Ignorance About Keynes’s Inexact, Approximation Approach to Measurement in the A Treatise on Probability in the Keynes-Tinbergen Exchanges of 1938–1940

There is no evidence that J. Tinbergen ever read the technical analysis provided by J. M. Keynes in Parts II and V of the A Treatise on Probability in the 1938-1940 time period or at any later time in his life. The same conclusion holds for econometricians in general. They simply do not understand what Keynes’s Inexact, Approximation Approach to Measurement in the A Treatise on Probability entails. Tinbergen and Keynes held diametrically opposed positions on measurement. Tinbergen’s approach to measurement was guided by the Limiting Frequency interpretation of probability while Keynes’s approach followed the logical approach of George Boole. Tinbergen’s physics background led him to deploy an exact approach to measurement based on the specification of probability distributions, like the normal and lognormal, with exact and precisely defined measurements. All probabilities were assumed to be well defined, precise, exact, determinate, definite, additive, linear, independent single number of answers. Keynes’s approach was an inexact one to measurement based on his modification of Boole’s original approach in 1854 in The Laws of Thought in chapters 16-21. Probabilities for Keynes were, with a few exceptions, partially defined, imprecise, inexact, indefinite, indeterminate, nonadditive, nonlinear, and dependent. Probability estimates for Keynes required two numbers to specify the probability within a lower and upper bound(limit), and nor one like Tinbergen’s approach. Keynes called this approach Approximation. Keynesian probabilities are interval-valued. There are no more than a single handful of academics, worldwide today, who grasp the fact that Keynes’s approach to measurement in the A Treatise on Probability was an interval-valued approach that was covered by Keynes in chapters III and XV. There was no economist alive in the late 1930’s-1940’s who understood Keynes’s approach in the A Treatise on Probability. An example of this ignorance is Lawrence Klein’s review in 1951 of Harrod’s biography of Keynes. The assessments of the Keynes –Tinbergen exchanges in 1938-40 were practically identical to the assessment made of the A Treatise on Probability by Ronald Fisher in his 1923 Eugenics Review of the A Treatise on Probability, where Fisher completely overlooked the inexact approach to measurement that Keynes presented in Parts I –V of the A Treatise on Probability. He dismissed the A Treatise on Probability based on his belief that Keynes was arguing that probability could not be applied since it was, at best, ordinal in nature. Only one economist in the 20th century understood exactly what it was that Keynes had done in the A Treatise on Probability. That economist was named F. Y. Edgeworth. However, Edgeworth died in 1926. His two 1922 book reviews of the A Treatise on Probability required a great deal of mathematical and statistical competence on the part of a reader. That mathematical competence was missing from the economics profession in the late 1930s and 1940s. The result was that Edgeworth’s two superb reviews were forgotten and never read by economists. The problem, from Keynes’s perspective, was that Tinbergen was trying to apply to economic data techniques which were only sound in physics, where laboratory controlled environments with detailed experimental design could generate data and replicate/duplicate the experiments worldwide. Keynes had always argued that economics was not a physical or life science like physics, engineering, biology or chemistry and that it could never be like physics. A great abyss thus separated Tinbergen from Keynes from the very start of the private exchange of letters that began in 1938 to the final exchanges in the 1939-40 time period in the Economic Journal. Tinbergen’s complete and total ignorance of Keynes’s approach to inexact measurement meant that the possibility of some type of intellectual compromise was impossible from the very start. The main reason for the completely divergent views expressed by Keynes and Tinbergen in their 1938-40 exchanges in the Economic Journal was due to their completely different technical backgrounds. Keynes’s background was logic, mathematics, and philosophy. Tinbergen’s background was physics. There simply was no common ground between the two academics.

Keynes’s Theory of Measurement is contained in Chapter III of Part I and in Chapter XV of Part II of the A Treatise on Probability (1921;1973 CWJMK Edition): Keynes Stated That the Exposition in Chapter III of the a Treatise on Probability Was 'Brief', While the Exposition in Chapter XV, Part II, Of the a Treatise on Probability, Was 'Detailed'

Professor Yasuhiro Sakai (see 2016; 2018) has argued that there is an mysterious problem in the A Treatise on Probability, 1921 in chapter 3 on page 39 (page 42 of the 1973 CWJMK edition). He argues that there is an unsolved mystery that involves this diagram that has remained unexplained in the literature. The mystery is that Keynes does not explain what he is doing in the analysis involving the diagram starting on the lower half of page 38 and ending on page 40 of chapter III. In fact, the mystery is solved for any reader of the A Treatise on Probability who reads page 37 and the upper half of page 38 carefully and closely. Keynes explicitly states on those pages that he will give only a brief discussion of the results of his approach to measurement on pages 38-40, but will provide a detailed discussion of his approach to measurement in Part II, after which the brief discussion of the results presented on pp.38-40 will be strengthened. The Post Keynesian (Joan Robinson, G L S Shackle, Sydney Weintraub, Paul Davidson) and Fundamentalist (Donald Moggridge, Robert Skidelsky, Gay Meeks, Anna Carabelli, Athol Fitzgibbons, Rod O’Donnell, Tony Lawson, Jochen Runde) schools of economics, as well as economists, in general, such as Jan Tinbergen and Lawrence Klein, have ignored chapter XV of the A Treatise on Probability. Keynes demonstrates on pp.161-163 of the A Treatise on Probability in chapter XV that his approach to measurement is an inexact approach to measurement using approximation to define interval valued probability, which is based on the upper-lower probabilities approach of George Boole, who discussed this approach in great detail in chapters 16-21 of his 1854 The Laws of Thought. Therefore, the only conclusion possible is that the “mysterious” diagram presented on page 39 of the A Treatise on Probability is an illustration of Keynes’s approximation technique using interval valued probability, since the problem on pages 162-163 of the A Treatise on Probability explicitly works with seven “non numerical” probabilities while the illustration of Keynes’s approach using the diagram on page 39 works with six “non numerical” probabilities and one numerical. It is impossible for the diagram on page 39 to support any claim, as has been done repeatedly for the last 45 years by the Post Keynesian and Keynesian. Fundamentalist schools, that Keynes’s theory was an ordinal theory that could only be applied some of the time. This leads precisely to the wrong conclusion that Keynes was arguing that macroeconomic measurement, in general, was impossible in economics, which was G L S Shackle’s conclusion. An understanding of chapter XV of the A Treatise on Probability explains the conflict that existed between J M Keynes and J Tinbergen on the pages of the Economic Journal of 1939 -1940.The major point of discussion, underlying all of Keynes’s major points, was that Tinbergen’s exact measurement approach, taken from macroscopic physics, using the Normal probability distribution’s precise, exact, definite, linear, additive, and independent probabilities, was not possible given the type of data available in macroeconomics. Only an inexact approach to measurement using imprecise and indeterminate interval valued probability was tenable. An understanding of chapter XV of Part II of the TP explains the fundamental point of disagreement between J M Keynes and J Tinbergen over the issue of measurement. Tinbergen brought his physic background with him to the study of economics. Tinbergen believed that the exact measurement approach that he had absorbed in his study of statistical physics, using additive, linear, exact, precise definite probability distributions like the Normal or log normal, could be used in the study of macroeconomics that would provide a precise and exact explanation of business cycles. Keynes, of course, given his great, overall experience in academia, industry, business, government, the stock markets, bond markets, money markets, banking, finance, and commodity futures markets, had vast experience that Tinbergen, an academic only, did not have. Keynes saw that Tinbergen’s application was the wrong one, although the technique would be applicable to studies of consumption and inventories.

How Keynes Solved the ‘Mystery’ of the Diagram on Page 39 (Page 42 of the 1973 CWJMK Edition) of the A Treatise on Probability in Part II in Chapter 15 on pp.161–163 Just As He Had Foretold on pp. 37–38 of Chapter III

On pp.37-38 of the A Treatise on Probability ,Keynes was very clear that he was only going to present a 'brief' analysis of non numerical probability graphically as an illustration. However, in Part II, he stated he would present a 'detailed' analysis. This detailed mathematical and logical analysis involved the application of his modified approach to inexact measurement using the method of approximation that had first been used by Boole in 1854. Keynes demonstrated and presented a simple, illustrated version of his interval valued approach in Chapter III of the A Treatise on Probability. In Chapter 15, he provided the detailed mathematical application of his logical theory of probability. Keynes followed through on his promise to demonstrate a purely mathematical treatment of non numerical probabilities.The real mystery is the complete and total failure of the Post Keynesians and Keynesian Fundamentalists to grasp Keynes’s completely clear analysis 98 YEARS after the publication of the A Treatise on Probability in 1921. So Keynes actually did precisely what he said he would do. The only real remaining mystery is why no philosopher, mathematician, or economist in the 20th or 21st centuries ever read chapters 15, 16, and 17 of the A Treatise on Probability. It has been 98 years since Keynes published his book in 1921. The time has come for serious scholars to move beyond Keynes’s introduction to measurement in chapter III of the A Treatise on Probability, using diagrams, to Keynes’s full scale mathematical and logical application of his inexact measurement approach that he called approximation based on Boole’s upper and lower, interval valued, estimate approach to probability and decision making.

Weighted cross-gradient function for joint inversion with the application to regional 3-D gravity and magnetic anomalies

In the absence of quantitative relationships, a cross-gradient function can be used to correlate unknown physical properties in a joint inversion of geophysical data sets. It introduces a structural correlation between properties. A commonly used, inexact approach adds a weighted cross-gradient term as a penalty to the cost function being minimized during inversion. This weighting factor needs to be tuned to balance the regularization and cross-gradient terms. In this paper we propose nonlinear weighting for the cross-gradient function which addresses the very different magnitudes of the cross-gradient and regularization terms. This approach also couples the weighting factors for the regularization and correlation terms reducing the number of tuning parameters. The approach is investigated for a synthetic case. Results are also shown for the 3-D joint inversion of high-resolution magnetic and gravity anomaly data from Southern Queensland in Australia with over 30 million cells.

J. M. Keynes (Inexact Measurement, Approximation, Non- Linearity, Non-Additivity, Interdependence, Imprecision) Versus J. Tinbergen (Exact Measurement, Linearity, Additivity, Independence, Precision) on Probability in 1939–1940: There Was No Middle Ground Between Them

J. M. Keynes (Inexact Measurement, Approximation, Non- Linearity, Non-Additivity, Interdependence, Imprecision) Versus J. Tinbergen (Exact Measurement, Linearity, Additivity, Independence, Precision) on Probability in 1939–1940: There Was No Middle Ground Between Them

Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai–Borwein gradient method

ABSTRACTAn adaptive nonmonotone spectral gradient method for the solution of distributed optimal control problem (OCP) for the viscous Burgers equation is presented in a black-box framework. Regarding the implicit function theorem, the OCP is transformed into an unconstrained nonlinear optimization problem (UNOP). For solving UNOP, an adaptive nonmonotone Barzilai–Borwein gradient method is proposed in which to make a globalization strategy, first an adaptive nonmonotone strategy which properly controls the degree of nonmonotonicity is presented and then is incorporated into an inexact line search approach to construct a more relaxed line search procedure. Also an adjoint technique is used to effectively evaluate the gradient. The low memory requirement and the guaranteed convergence property make the proposed method quite useful for large-scale OCPs. The efficiency of the presented method is supported by numerical experiments.

Preconditioning PDE-constrained optimization with [formula omitted]-sparsity and control constraints

PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including nonsmooth L1 sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L1-terms as well as additional inclusion of box control constraints require the use of semismooth Newton methods. We propose robust preconditioners for different formulations of the Newton equation. With the inclusion of a line-search strategy and an inexact approach for the solution of the linear systems, the resulting semismooth Newton’s method is reliable for practical problems. Our results are underpinned by a theoretical analysis of the preconditioned matrix. Numerical experiments illustrate the robustness of the proposed scheme.

Distributed Learning of Predictive Structures From Multiple Tasks Over Networks

This paper is concerned with the problem of distributed multitask learning over networks, which aims to simultaneously infer multiple node-specific parameter vectors in a collaborative manner. Most of the existing works on the distributed multitask problem modeled the task relatedness by assuming some similarities of parameter vectors in an explicit way. In this paper, we implicitly model the similarity of parameter vectors by assuming that the parameter vectors share a common low-dimensional predictive structure on hypothesis spaces, which is learned using the available data in networks. A distributed structure learning algorithm for the in-network cooperative estimation problem is then derived based on the block coordinate descent method integrated with the inexact alternating direction method of multipliers technique. Simulations on both synthetic and real-world datasets are given to verify the effectiveness of the proposed algorithm. In the case that each node shares a common predictive subspace, it is demonstrated that the proposed multitask algorithm outperforms the noncooperative learning algorithm. Moreover, the use of the inexact approach can significantly reduce the communication bandwidth and still provide the same optimal solution as the corresponding centralized approach.

A feasibility study of stateful automaton packet inspection for streaming application detection systems

ABSTRACTThe rapid development of the internet has brought huge benefits and social impacts; however, internet security has also become a great problem for users, since traditional approaches to packet classification cannot achieve satisfactory detection performance due to their low accuracy and efficiency. In this paper, a new stateful packet inspection method is introduced, which can be embedded in the network gateway and used by a streaming application detection system. This new detection method leverages the inexact automaton approach, using part of the header field and part of the application layer data of a packet. Based on this approach, an advanced detection system is proposed for streaming applications. The workflow of the system involves two stages: the training stage and the detection stage. In the training stage, the system initially captures characteristic patterns from a set of application packet flows. After this training is completed, the detection stage allows the user to detect the target application by capturing new application flows. This new detection approach is also evaluated using experimental analysis; the results of this analysis show that this new approach not only simplifies the management of the state detection system, but also improves the accuracy of data flow detection, making it feasible for real-world network applications.

A heuristic for cumulative vehicle routing using column generation

Cumulative vehicle routing problems are a simplified model of fuel consumption in vehicle routing problems. Here we computationally study, an inexact approach for constructing solutions to cumulative vehicle routing problems based on rounding solutions to a linear program. The linear program is based on the set cover formulation and is solved using column generation. The pricing subproblem is solved heuristically using dynamic programming. Simulation results show that a simple scalable strategy gives solutions with cost close to the lower bound given by the linear programming relaxation. We also give theoretical bounds on the integrality gap of the set cover formulation.

Inexact uniformization and GMRES methods for large Markov chains

SUMMARYInexact algorithms allow certain operations (typically matrix–vector products or certain function evaluations) to be performed inexactly, either out of necessity or deliberately, in view of trading accuracy for speed. We review recent findings that show the impact of the inexact approach in the uniformization and generalized minimal residual (GMRES) methods when computing the transient, stationary, and cumulative solutions of realistic Markov chain problems of large size arising from computer systems and biochemical reactions. Copyright © 2011 John Wiley & Sons, Ltd.

A Comparison-Contrast of Adam Smith, JM Keynes and Jeremy Bentham on Probability, Risk, Uncertainty, Optimism-Pessimism and Decision Making with Applications Concerning Banking, Insurance and Speculation

Both Smith and Keynes have very similar conceptual approaches to what probability is, how it is used and applied and the areas of application in which it can aid a decision maker. They both accept an interval approach to probability based on inequalities and bounds versus ordinal, subjectivist Bayesian and relative frequency approaches. This led them to have very similar views with respect to analyzing speculative bubbles, lenders versus borrowers risk assessments, the dangers of speculative financeespecially if the bankers themselves are or become speculators and/or are financing speculators, and a policy of maintaining low, fixed rates of interest in order to control the speculative demand for money. Both Keynes and Smith showed how their very similar constructs led to the creation of a viable insurance industry that would be based on an inexact approach to probability. Bentham’s views on probability and decision making are directly opposed to those of Smith and Keynes. Bentham can be regarded as the founder of the subjectivist, Bayesian approach to decision making. The modern Subjective Expected Utility (SEU) approach can be traced back to Bentham’s original arguments about the ability of rational decision makers to calculate using precise numerical probabilities and outcomes. Bentham is the first, major proponent of the exact approach to probability and decision making. Smith and Keynes would reject the Kahneman-Tversky behavioral economist “Heuristics and Biases“ approach that regards mathematical probability as the normative criterion that all decision makers should attempt to emulate since this approach is a more advanced mathematical version of Bentham’s original approach. Mathematical probability, which requires the use of precise or sharp numerical probabilities, can only be normative in the case where the decision maker has a complete information set and/or knows for certain that a specific probability distribution will apply now and in the future. The mathematical laws of the probability calculus only hold as a limiting case whenever humans are part of the equation. Both Keynes and Smith rely on an inexact, interval, non additive, nonlinear approach that directly conflicts with the exact, single number, additive, linear approach recommended by Tversky and Kahneman as being the hallmark of human rationality in decision making. On the other hand, Bentham’s approach is an earlier version of the exact, linear and additive approach recommended as being rational by Tversky-Kahneman. Smith and Keynes, however, did emphasize fields that today are called Cognitive Science and Cognitive Psychology. Here pattern recognition, similarity, induction and intuition played an important role.

Evaluation of the performance of inexact GMRES

The inexact GMRES algorithm is a variant of the GMRES algorithm where matrix–vector products are performed inexactly, either out of necessity or deliberately, as part of a trading of accuracy for speed. Recent studies have shown that relaxing matrix–vector products in this way can be justified theoretically and experimentally. Research, so far, has focused on decreasing the workload per iteration without significantly affecting the accuracy. But relaxing the accuracy per iteration is liable to increase the number of iterations, thereby increasing the overall runtime, which could potentially end up being greater than that of the exact GMRES if there were not enough savings in the matrix–vector products. In this paper, we assess the benefit of the inexact approach in terms of actual CPU time derived from realistic problems, and we provide cases that provide instructive insights into results affected by the build-up of the inexactness. Such information is of vital importance to practitioners who need to decide whether switching their workflow to the inexact approach is worth the effort and the risk that might come with it. Our assessment is drawn from extensive numerical experiments that gauge the effectiveness of the inexact scheme and its suitability for use in addressing certain problems, depending on how much inexactness is allowed in the matrix–vector products.

Inexact fuzzy-stochastic mixed integer programming approach for long-term planning of waste management–––Part B: Case study

In this study, a solid waste decision-support system was developed for the long-term planning of waste management in the City of Regina, Canada. Interactions among various system components, objectives, and constraints will be analyzed. Issues concerning planning for cost-effective diversion and prolongation of the landfill will be addressed. Decisions of system-capacity expansion and waste allocation within a multi-facility, multi-option, and multi-period context will be obtained. The obtained results would provide useful information and decision-support for the City's solid waste management and planning. In the application, four scenarios are considered. Through the above scenario analyses under different waste-management policies, useful decision support for the City's solid waste managers and decision makers was generated. Analyses for the effects of varied policies (for allowable waste flows to different facilities) under 35 and 50% diversion goals were also undertaken. Tradeoffs among system cost and constraint-violation risk were analyzed. Generally, a policy with lower allowable waste-flow levels corresponded to a lower system cost under advantageous conditions but, at the same time, a higher penalty when such allowances were violated. A policy with higher allowable flow levels corresponded to a higher cost under disadvantageous conditions. The modeling results were useful for (i) scheduling adequate time and capacity for long-term planning of the facility development and/or expansion in the city's waste management system, (ii) adjusting of the existing waste flow allocation patterns to satisfy the city's diversion goal, and (iii) generating of desired policies for managing the city's waste generation, collection and disposal.