Search results for: Stochastic ordering.

Authors: Samia Alblwi, Amani Ayad

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Mutation testing is the art of generating syntactic variations of a base program and checking whether a candidate test suite can identify all the mutants that are not semantically equivalent to the base; this technique can be used to assess the quality of test suite. One of the main obstacles to the widespread use of mutation testing is cost, as even small programs (a few dozen lines of code) can give rise to a large number of mutants (up to hundreds); this has created an incentive to seek to reduce the number of mutants while preserving their collective effectiveness. Two criteria have been used to reduce the size of mutant sets: equivalence, which aims to partition the set of mutants into equivalence classes modulo semantic equivalence, and selecting one representative per class; and, subsumption, which aims to define a partial ordering among mutants that ranks mutants by effectiveness and seeks to select maximal elements in this ordering. In this paper, we analyze these two policies using analytical and empirical criteria.

Keywords: Mutation testing, mutant sets, mutant equivalence, mutant subsumption, mutant set minimization.

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Authors: Mizan Ahmed, Srikanth Venkatesan

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Geological and tectonic framework indicates that Bangladesh is one of the most seismically active regions in the world. The Bengal Basin is at the junction of three major interacting plates: the Indian, Eurasian, and Burma Plates. Besides there are many active faults within the region, e.g. the large Dauki fault in the north. The country has experienced a number of destructive earthquakes due to the movement of these active faults. Current seismic provisions of Bangladesh are mostly based on earthquake data prior to the 1990. Given the record of earthquakes post 1990, there is a need to revisit the design provisions of the code. This paper compares the base shear demand of three major cities in Bangladesh: Dhaka (the capital city), Sylhet, and Chittagong for earthquake scenarios of magnitudes 7.0MW, 7.5MW, 8.0MW, and 8.5MW using a stochastic model. In particular, the stochastic model allows the flexibility to input region specific parameters such as shear wave velocity profile (that were developed from Global Crustal Model CRUST2.0) and include the effects of attenuation as individual components. Effects of soil amplification were analysed using the Extended Component Attenuation Model (ECAM). Results show that the estimated base shear demand is higher in comparison with code provisions leading to the suggestion of additional seismic design consideration in the study regions.

Keywords: Attenuation, earthquake, ground motion, stochastic, seismic hazard.

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Authors: Myat Su Mon Win

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The pseudorandom number generators based on linear feedback shift registers (LFSRs), are very quick, easy and secure in the implementation of hardware and software. Thus they are very popular and widely used. But LFSRs lead to fairly easy cryptanalysis due to their completely linearity properties. In this paper, we propose a stochastic generator, which is called Random Feedback Shift Register (RFSR), using stochastic transformation (Random block) with one-way and non-linearity properties.

Keywords: Linear Feedback Shift Register, Non Linearity, R_Block, Random Feedback Shift Register

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Authors: Turgay Temel, John Hallam

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A recent neurospiking coding scheme for feature extraction from biosonar echoes of various plants is examined with avariety of stochastic classifiers. Feature vectors derived are employedin well-known stochastic classifiers, including nearest-neighborhood,single Gaussian and a Gaussian mixture with EM optimization.Classifiers' performances are evaluated by using cross-validation and bootstrapping techniques. It is shown that the various classifers perform equivalently and that the modified preprocessing configuration yields considerably improved results.

Keywords: Classification, neuro-spike coding, non-parametricmodel, parametric model, Gaussian mixture, EM algorithm.

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Authors: S. Chadsuthi, W. Triampo, C. Modchang, P. Kanthang, D. Triampo, N. Nuttavut

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We present analysis of spatial patterns of generic disease spread simulated by a stochastic long-range correlation SIR model, where individuals can be infected at long distance in a power law distribution. We integrated various tools, namely perimeter, circularity, fractal dimension, and aggregation index to characterize and investigate spatial pattern formations. Our primary goal was to understand for a given model of interest which tool has an advantage over the other and to what extent. We found that perimeter and circularity give information only for a case of strong correlation– while the fractal dimension and aggregation index exhibit the growth rule of pattern formation, depending on the degree of the correlation exponent (β). The aggregation index method used as an alternative method to describe the degree of pathogenic ratio (α). This study may provide a useful approach to characterize and analyze the pattern formation of epidemic spreading

Keywords: spatial pattern epidemics, aggregation index, fractaldimension, stochastic, long-rang epidemics

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Authors: Xiang Zhang, David Rey, S. Travis Waller

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Stochastic User Equilibrium (SUE) model is a widely used traffic assignment model in transportation planning, which is regarded more advanced than Deterministic User Equilibrium (DUE) model. However, a problem exists that the performance of the SUE model depends on its error term parameter. The objective of this paper is to propose a systematic method of determining the appropriate error term parameter value for the SUE model. First, the significance of the parameter is explored through a numerical example. Second, the parameter calibration method is developed based on the Logit-based route choice model. The calibration process is realized through multiple nonlinear regression, using sequential quadratic programming combined with least square method. Finally, case analysis is conducted to demonstrate the application of the calibration process and validate the better performance of the SUE model calibrated by the proposed method compared to the SUE models under other parameter values and the DUE model.

Keywords: Parameter calibration, sequential quadratic programming, Stochastic User Equilibrium, traffic assignment, transportation planning.

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Authors: Yen-Po Wang, Ming-Chih Huang, Ming-Lian Chang

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A scheme integrated with deterministic–stochastic subspace system identification and the method of damage localization vector is proposed in this study for damage detection of structures based on seismic response data. A series of shaking table tests using a five-storey steel frame has been conducted in National Center for Research on Earthquake Engineering (NCREE), Taiwan. Damage condition is simulated by reducing the cross-sectional area of some of the columns at the bottom. Both single and combinations of multiple damage conditions at various locations have been considered. In the system identification analysis, either full or partial observation conditions have been taken into account. It has been shown that the damaged stories can be identified from global responses of the structure to earthquakes if sufficiently observed. In addition to detecting damage(s) with respect to the intact structure, identification of new or extended damages of the as-damaged (ill-conditioned) counterpart has also been studied. The proposed scheme proves to be effective.

Keywords: Damage locating vectors, deterministic-stochastic subspace system, shaking table tests, system identification.

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Authors: Xianqing Liu, Shouming Zhong, Lijiang Xiang

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In this paper, a stochastic predator-prey system with Bedding-DeAngelis functional response is studied. By constructing a suitable Lyapunov founction, sufficient conditions for species to be stochastically permanent is established. Meanwhile, we show that the species will become extinct with probability one if the noise is sufficiently large.

Keywords: Stochastically permanent, extinct, white noise, Bedding-DeAngelis functional response.

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Authors: Jihad S. Daba, J. P. Dubois

Abstract:

Fading noise degrades the performance of cellular communication, most notably in femto- and pico-cells in 3G and 4G systems. When the wireless channel consists of a small number of scattering paths, the statistics of fading noise is not analytically tractable and poses a serious challenge to developing closed canonical forms that can be analysed and used in the design of efficient and optimal receivers. In this context, noise is multiplicative and is referred to as stochastically local fading. In many analytical investigation of multiplicative noise, the exponential or Gamma statistics are invoked. More recent advances by the author of this paper utilized a Poisson modulated-weighted generalized Laguerre polynomials with controlling parameters and uncorrelated noise assumptions. In this paper, we investigate the statistics of multidiversity stochastically local area fading channel when the channel consists of randomly distributed Rayleigh and Rician scattering centers with a coherent Nakagami-distributed line of sight component and an underlying doubly stochastic Poisson process driven by a lognormal intensity. These combined statistics form a unifying triply stochastic filtered marked Poisson point process model.

Keywords: Cellular communication, femto- and pico-cells, stochastically local area fading channel, triply stochastic filtered marked Poisson point process.

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Authors: Athanasios T. Alexiou, Maria M. Psiha, John A. Rekkas, Panayiotis M. Vlamos

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Mitochondria are dynamic organelles, capable to interact with each other. While the number of mitochondria in a cell varies, their quality and functionality depends on the operation of fusion, fission, motility and mitophagy. Nowadays, several researches declare as an important factor in neurogenerative diseases the disruptions in the regulation of mitochondrial dynamics. In this paper a stochastic model in BioAmbients calculus is presented, concerning mitochondrial fusion and its distribution in the renewal of mitochondrial population in a cell. This model describes the successive and dependent stages of protein synthesis, protein-s activation and merging of two independent mitochondria.

Keywords: Mitochondrial Dynamics, P-Calculus, StochasticModeling.

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Authors: T. T. Manikandan, Rajeev Sukumaran

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Coral Reefs are very important for the majority of marine ecosystems. But, such vital species are under major threat due to the factors such as ocean acidification, overfishing, and coral bleaching. To conserve the coral reefs, reef restoration activities are carried out across the world. After reef restoration, various parameters have to be monitored in order to ensure the overall effectiveness of the reef restoration. Underwater Wireless Sensor Network (UWSN) based  monitoring is widely adopted for such long monitoring activities. Since monitoring of coral reef restoration activities is time sensitive, the QoS guarantee offered by the network with respect to delay is vital. So this research focuses on the analytical modeling of network layer delay using Stochastic Network Calculus (SNC). The core focus of the proposed model will be on the analysis of stochastic dependencies between the network flow and deriving the stochastic delay bounds for the flows that traverse in tandem in UWSNs. The derived analytical bounds are evaluated for their effectiveness using discrete event simulations.

Keywords: Coral Reef Restoration, SNC, SFA, PMOO, Tandem of Queues, Delay Bound.

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Authors: Zixin Liu, Shu Lü, Shouming Zhong, Mao Ye

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This paper studies the pth moment exponential synchronization of a class of stochastic neural networks with mixed delays. Based on Lyapunov stability theory, by establishing a new integrodifferential inequality with mixed delays, several sufficient conditions have been derived to ensure the pth moment exponential stability for the error system. The criteria extend and improve some earlier results. One numerical example is presented to illustrate the validity of the main results.

Keywords: pth Moment Exponential synchronization, Stochastic, Neural networks, Mixed time delays

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Authors: Davod Khojasteh Salkuyeh

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An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.

Keywords: Ordinary differential equations, optimal stepsize, error, stochastic arithmetic, CESTAC, CADNA.

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Authors: Siliang Wang, Minghui Wang, Jun Hu

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A novel path planning approach is presented to solve optimal path in stochastic, time-varying networks under priori traffic information. Most existing studies make use of dynamic programming to find optimal path. However, those methods are proved to be unable to obtain global optimal value, moreover, how to design efficient algorithms is also another challenge. This paper employs a decision theoretic framework for defining optimal path: for a given source S and destination D in urban transit network, we seek an S - D path of lowest expected travel time where its link travel times are discrete random variables. To solve deficiency caused by the methods of dynamic programming, such as curse of dimensionality and violation of optimal principle, an integer programming model is built to realize assignment of discrete travel time variables to arcs. Simultaneously, pruning techniques are also applied to reduce computation complexity in the algorithm. The final experiments show the feasibility of the novel approach.

Keywords: pruning method, stochastic, time-varying networks, optimal path planning.

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Authors: Mohammad Mahdi Paydar, Armin Cheraghalipour, Mostafa Hajiaghaei-Keshteli

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Nowadays, in advanced countries, agriculture as one of the most significant sectors of the economy, plays an important role in its political and economic independence. Due to farmers' lack of information about products' demand and lack of proper planning for harvest time, annually the considerable amount of products is corrupted. Besides, in this paper, we attempt to improve these unfavorable conditions via designing an effective supply chain network that tries to minimize total costs of agricultural products along with minimizing shortage in demand points. To validate the proposed model, a stochastic optimization approach by using a branch and bound solver of the LINGO software is utilized. Furthermore, to accumulate the data of parameters, a case study in Mazandaran province placed in the north of Iran has been applied. Finally, using ɛ-constraint approach, a Pareto front is obtained and one of its Pareto solutions as best solution is selected. Then, related results of this solution are explained. Finally, conclusions and suggestions for the future research are presented.

Keywords: Perishable products, stochastic optimization, agricultural supply chain, ɛ-constraint.

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Authors: Zdravka Aljinović, Branka Marasović, Blanka Škrabić

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The paper provides a discussion of the most relevant aspects of yield curve modeling. Two classes of models are considered: stochastic and parsimonious function based, through the approaches developed by Vasicek (1977) and Nelson and Siegel (1987). Yield curve estimates for Croatia are presented and their dynamics analyzed and finally, a comparative analysis of models is conducted.

Keywords: the term structure of interest rates, Vasicek model, Nelson-Siegel model, Croatian Government market.

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Authors: Hassan Manouzi

Abstract:

We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.

Keywords: Least squares, Wick product, SPDEs, finite element, Wiener chaos expansion, gradient method.

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Authors: Bassem Roushdy, Nahed Sobhy, Abdelrhim Abdelhamid, Ahmed Mahmoud

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Most papers model Joint Replenishment Problem (JRP) as a (kT,S) where kT is a multiple value for a common review period T,and S is a predefined order up to level. In general the (T,S) policy is characterized by a long out of control period which requires a large amount of safety stock compared to the (R,Q) policy. In this paper a probabilistic model is built where an item, call it item(i), with the shortest order time between interval (T)is modeled under (R,Q) policy and its inventory is continuously reviewed, while the rest of items (j) are periodically reviewed at a definite time corresponding to item

Keywords: Inventory management, Joint replenishment, policy evaluation, stochastic process

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Authors: Jihad S. Daba (Jean-Pierre Dubois), Philip Jreije

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Speckled images arise when coherent microwave, optical, and acoustic imaging techniques are used to image an object, surface or scene. Examples of coherent imaging systems include synthetic aperture radar, laser imaging systems, imaging sonar systems, and medical ultrasound systems. Speckle noise is a form of object or target induced noise that results when the surface of the object is Rayleigh rough compared to the wavelength of the illuminating radiation. Detection and estimation in images corrupted by speckle noise is complicated by the nature of the noise and is not as straightforward as detection and estimation in additive noise. In this work, we derive stochastic models for speckle noise, with an emphasis on speckle as it arises in medical ultrasound images. The motivation for this work is the problem of segmentation and tissue classification using ultrasound imaging. Modeling of speckle in this context involves partially developed speckle model where an underlying Poisson point process modulates a Gram-Charlier series of Laguerre weighted exponential functions, resulting in a doubly stochastic filtered Poisson point process. The statistical distribution of partially developed speckle is derived in a closed canonical form. It is observed that as the mean number of scatterers in a resolution cell is increased, the probability density function approaches an exponential distribution. This is consistent with fully developed speckle noise as demonstrated by the Central Limit theorem.

Keywords: Doubly stochastic filtered process, Poisson point process, segmentation, speckle, ultrasound

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Authors: Karolína Vozková, Matěj Kuc

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This paper analyzes recent trends in cost efficiency of European cooperative banks using efficient frontier analysis. Our methodology is based on stochastic frontier analysis which is run on a set of 649 European cooperative banks using data between 2006 and 2015. Our results show that average inefficiency of European cooperative banks is increasing since 2008, smaller cooperative banks are significantly more efficient than the bigger ones over the whole time period and that share of net fee and commission income to total income surprisingly seems to have no impact on bank cost efficiency.

Keywords: Cooperative banks, cost efficiency, efficient frontier analysis, stochastic frontier analysis, net fee and commission income.

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Authors: Rosita Suhaimi, Firdaus Abdullah, Chong Fen Nee, Nurhani Aba Ibrahim

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This paper attempts to identify the significance of Information and Communications Technology (ICT) and competitiveness to the profit efficiency of commercial banks in Malaysia. The profit efficiency of commercial banks in Malaysia, the dependent variable, was estimated using the Stochastic Frontier Approach (SFA) on a sample of unbalanced panel data, covering 23 commercial banks, between 1995 to 2007. Based on the empirical results, ICT was not found to exert a significant impact on profit efficiency, whereas competitiveness, non ICT stock expenditure and ownership were significant contributors. On the other hand, the size of banks was found to have significantly reduced profit efficiency, opening up for various interpretations of the interrelated role of ICT and competition.

Keywords: Competitiveness, Profit Efficiency, Stochastic Frontier Analysis

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Authors: Karim Hamidi Machekposhti, Hossein Sedghi

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This study was designed to find the best stochastic model (using of time series analysis) for annual extreme streamflow (peak and maximum streamflow) of Karkheh River at Iran. The Auto-regressive Integrated Moving Average (ARIMA) model used to simulate these series and forecast those in future. For the analysis, annual extreme streamflow data of Jelogir Majin station (above of Karkheh dam reservoir) for the years 1958–2005 were used. A visual inspection of the time plot gives a little increasing trend; therefore, series is not stationary. The stationarity observed in Auto-Correlation Function (ACF) and Partial Auto-Correlation Function (PACF) plots of annual extreme streamflow was removed using first order differencing (d=1) in order to the development of the ARIMA model. Interestingly, the ARIMA(4,1,1) model developed was found to be most suitable for simulating annual extreme streamflow for Karkheh River. The model was found to be appropriate to forecast ten years of annual extreme streamflow and assist decision makers to establish priorities for water demand. The Statistical Analysis System (SAS) and Statistical Package for the Social Sciences (SPSS) codes were used to determinate of the best model for this series.

Keywords: Stochastic models, ARIMA, extreme streamflow, Karkheh River.

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Authors: Salinee Thumronglaohapun

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The proper number and appropriate locations of service centers can save cost, raise revenue and gain more satisfaction from customers. Establishing service centers is high-cost and difficult to relocate. In long-term planning periods, several factors may affect the service. One of the most critical factors is uncertain demand of customers. The opened service centers need to be capable of serving customers and making a profit although the demand in each period is changed. In this work, the capacitated location-allocation problem with stochastic demand is considered. A mathematical model is formulated to determine suitable locations of service centers and their allocation to maximize total profit for multiple planning periods. Two heuristic methods, a local search and genetic algorithm, are used to solve this problem. For the local search, five different chances to choose each type of moves are applied. For the genetic algorithm, three different replacement strategies are considered. The results of applying each method to solve numerical examples are compared. Both methods reach to the same best found solution in most examples but the genetic algorithm provides better solutions in some cases.

Keywords: Location-allocation problem, stochastic demand, local search, genetic algorithm.

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Authors: Y. Long, L. Liu, K. V. Branin

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One challenge faced by procurement decision-maker during the acquisition process is how to compare similar products from different suppliers and allocate orders among different products or services. This work focuses on allocating orders among multiple suppliers considering rebate. The objective function is to minimize the total acquisition cost including purchasing cost and rebate benefit. Rebate benefit is complex and difficult to estimate at the ordering step. Rebate rules vary for different suppliers and usually change over time. In this work, we developed a system to collect the rebate policies, standardized the rebate policies and developed two-stage optimization models for ordering allocation. Rebate policy with multi-tiers is considered in modeling. The discontinuous cost function of rebate benefit is formulated for different scenarios. A piecewise linear function is used to approximate the discontinuous cost function of rebate benefit. And a Mixed Integer Programing (MIP) model is built for order allocation problem with multi-tier rebate. A case study is presented and it shows that our optimization model can reduce the total acquisition cost by considering rebate rules.

Keywords: Discontinuous cost function, mixed integer programming, optimization, procurement, rebate.

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Authors: Irwin Ma, Tony Wong, Thiagas Sankar

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By systematically applying different engineering methods, difficult financial problems become approachable. Using a combination of theory and techniques such as wavelet transform, time series data mining, Markov chain based discrete stochastic optimization, and evolutionary algorithms, this work formulated a strategy to characterize and forecast non-linear time series. It attempted to extract typical features from the volatility data sets of S&P100 and S&P500 indices that include abrupt drops, jumps and other non-linearity. As a result, accuracy of forecasting has reached an average of over 75% surpassing any other publicly available results on the forecast of any financial index.

Keywords: Discrete stochastic optimization, genetic algorithms, genetic programming, volatility forecast

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Authors: Changzhao Li, Juan Zhang

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In this paper, we investigate a class of fuzzy Cohen- Grossberg neural networks with time delays and impulsive effects. By virtue of stochastic analysis, Halanay inequality for stochastic differential equations, we find sufficient conditions for the global exponential square-mean synchronization of the FCGNNs under noise perturbation. In particular, the traditional assumption on the differentiability of the time-varying delays is no longer needed. Finally, a numerical example is given to show the effectiveness of the results in this paper.

Keywords: Fuzzy Cohen-Grossberg neural networks (FCGNNs), complete synchronization, time delays, impulsive, noise perturbation.

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Authors: András Király, Tamás Varga, János Abonyi

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Since supply chains highly impact the financial performance of companies, it is important to optimize and analyze their Key Performance Indicators (KPI). The synergistic combination of Particle Swarm Optimization (PSO) and Monte Carlo simulation is applied to determine the optimal reorder point of warehouses in supply chains. The goal of the optimization is the minimization of the objective function calculated as the linear combination of holding and order costs. The required values of service levels of the warehouses represent non-linear constraints in the PSO. The results illustrate that the developed stochastic simulator and optimization tool is flexible enough to handle complex situations.

Keywords: stochastic processes, empirical distributions, Monte Carlo simulation, PSO, supply chain management

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Authors: Paola Lecca, Lorenzo Dematte

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Reactiondiffusion systems are mathematical models that describe how the concentration of one or more substances distributed in space changes under the influence of local chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose general form is ÔêétX(x, t) = DΔX(x, t), where X(x, t) is the state vector, D is the matrix of the diffusion coefficients and Δ is the Laplace operator. If the solute move in an homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and of solutes and on local temperature of the medium. In this paper a new stochastic reaction-diffusion model in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces of solvent and solute is presented. Such a model provides a more realistic description of the molecular kinetics in non-homogenoeus and highly structured media as the intra- and inter-cellular spaces. The movement of a molecule A from a region i to a region j of the space is described as a first order reaction Ai k- → Aj , where the rate constant k depends on the diffusion coefficient. Representing the diffusional motion as a chemical reaction allows to assimilate a reaction-diffusion system to a pure reaction system and to simulate it with Gillespie-inspired stochastic simulation algorithms. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the specific speed of reaction and diffusion events. Redi is the software tool, developed to implement the model of reaction-diffusion kinetics and dynamics. It is a free software, that can be downloaded from http://www.cosbi.eu. To demonstrate the validity of the new reaction-diffusion model, the simulation results of the chaperone-assisted protein folding in cytoplasm obtained with Redi are reported. This case study is redrawing the attention of the scientific community due to current interests on protein aggregation as a potential cause for neurodegenerative diseases.

Keywords: Reaction-diffusion systems, Fick's law, stochastic simulation algorithm.

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Authors: Ho Yuan-Hong, Hunag Chiung-Ju

Abstract:

This study conducts simulation analyses to find the optimal debt ceiling of Taiwan, while factoring in welfare maximization under a dynamic stochastic general equilibrium framework. The simulation is based on Taiwan's 2001 to 2011 economic data and shows that welfare is maximized at a debt/GDP ratio of 0.2, increases in the debt/GDP ratio leads to increases in both tax and interest rates and decreases in the consumption ratio and working hours. The study results indicate that the optimal debt ceiling of Taiwan is 20% of GDP, where if the debt/GDP ratio is greater than 40%, the welfare will be negative and result in welfare loss.

Keywords: Debt sustainability, optimal debt ceiling, dynamic stochastic general equilibrium, welfare maximization.

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Authors: Akbar Banitalebi, Mohd Ismail Abd Aziz, Rohanin Ahmad

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A new stochastic algorithm called Probabilistic Global Search Johor (PGSJ) has recently been established for global optimization of nonconvex real valued problems on finite dimensional Euclidean space. In this paper we present convergence guarantee for this algorithm in probabilistic sense without imposing any more condition. Then, we jointly utilize this algorithm along with control parameterization technique for the solution of constrained optimal control problem. The numerical simulations are also included to illustrate the efficiency and effectiveness of the PGSJ algorithm in the solution of control problems.

Keywords: Optimal Control Problem, Constraints, Direct Methods, Stochastic Algorithm

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