Form Of Mathematical Expectation Research Articles

Spectral projected subgradient method for nonsmooth convex optimization problems

We consider constrained optimization problems with a nonsmooth objective function in the form of mathematical expectation. The Sample Average Approximation (SAA) is used to estimate the objective function and variable sample size strategy is employed. The proposed algorithm combines an SAA subgradient with the spectral coefficient in order to provide a suitable direction which improves the performance of the first order method as shown by numerical results. The step sizes are chosen from the predefined interval and the almost sure convergence of the method is proved under the standard assumptions in stochastic environment. To enhance the performance of the proposed algorithm, we further specify the choice of the step size by introducing an Armijo-like procedure adapted to this framework. Considering the computational cost on machine learning problems, we conclude that the line search improves the performance significantly. Numerical experiments conducted on finite sum problems also reveal that the variable sample strategy outperforms the full sample approach.

On some forms of registration and display of modes of operation of vehicles

The article discusses the shortcomings of existing methods of accounting for engine operating time for the purpose of forming standards and modes of maintenance and repair. The results of testing a vehicle using specialized tools, including an adapter to the OBD diagnostic connector and a smartphone with special software, which allows registering, storing and accumulating data on the speed and load modes of the engine and the car, are presented. According to the test results, the drawbacks of registering the parameters of speed and load modes are analyzed by traditional characteristics in the form of mathematical expectation and variance of random variables. Methods of formalized display of the character of parameter changes in time are considered using autocorrelation functions and matrices of transition probabilities. The results of testing the analysis of the nature of changes in operating parameters in urban and mainline operating conditions are presented.

Penalty variable sample size method for solving optimization problems with equality constraints in a form of mathematical expectation

Equality-constrained optimization problems with deterministic objective function and constraints in the form of mathematical expectation are considered. The constraints are approximated by employing the sample average where the sample size varies throughout the iterations in an adaptive manner. The proposed method incorporates variable sample size scheme with cumulative and unbounded sample into the well- known quadratic penalty iterative procedure. Line search is used for globalization and the sample size is updated in a such way to preserve the balance between two types of errors—errors coming from the sample average approximation and the approximation of the optimal point. Moreover, the penalty parameter is also updated in an adaptive way. We prove that the proposed algorithm pushes the sample size and the penalty parameter to infinity which further allows us to prove the almost sure convergence towards a Karush-Kuhn-Tucker optimal point of the original problem under the rather standard assumptions. Numerical comparison on a set of relevant problems shows the advantage of the proposed adaptive scheme over the heuristic (predetermined) sample scheduling in terms of number of function evaluations as a measure of the optimization cost.

Spectral projected gradient method for stochastic optimization

We consider the Spectral Projected Gradient method for solving constrained optimization problems with the objective function in the form of mathematical expectation. It is assumed that the feasible set is convex, closed and easy to project on. The objective function is approximated by a sequence of different Sample Average Approximation functions with different sample sizes. The sample size update is based on two error estimates—SAA error and approximate solution error. The Spectral Projected Gradient method combined with a nonmonotone line search is used. The almost sure convergence results are achieved without imposing explicit sample growth condition. Preliminary numerical results show the efficiency of the proposed method.

Variable sample size method for equality constrained optimization problems

An equality constrained optimization problem with a deterministic objective function and constraints in the form of mathematical expectation is considered. The constraints are transformed into the Sample Average Approximation form resulting in deterministic problem. A method which combines a variable sample size procedure with line search is applied to a penalty reformulation. The method generates a sequence that converges towards first-order critical points. The final stage of the optimization procedure employs the full sample and the SAA problem is eventually solved with significantly smaller cost. Preliminary numerical results show that the proposed method can produce significant savings compared to SAA method and some heuristic sample update counterparts while generating a solution of the same quality.

Forecasting time series of the market indicators based on a nonlinear autoregressive neural network

The modern practice of economic research relies heavily on mathematical models that make it possible to reveal hidden regularities in statistical data and make forecasts on their basis. Linear models based on vector autoregression (VAR) are the most common. However, the relationship between time series in the economy is often difficult to identify, so non-linear autoregressive (NAR) models show more reliable results. Artificial neural networks (ANNs) are usually used for implementation of these models, but ANNs do not provide the possibility of estimating the forecast in the form of mathematical expectation and a standard deviation. Therefore, the model proposed in the article combines two blocks: VAR and NAR. NAR is used to construct a prediction for a given number of points, and VAR for estimating the forecast in the form of a mathematical expectation and a standard deviation. The evaluation of the model was carried out on the daily data: exchange rate USD / RUB and “Brent” oil from 1.01.2016 to 1.03.2017. The average accuracy of forecasting the trend for the dollar was 54.9%, for the oil prices it was 54.0%. In this case, the relative error in predicting the dollar rate was from 1.09% (for the first point) to 2.01% (for the tenth point); the relative error in forecasting oil prices was from 1.28% (for the first point) to 4.58 % (for the tenth point). Thus, the model showed accurate results when predicting dynamic series and can be used to solve other forecasting problems. In particular, it is expedient to use the model as one of the factors when making investment decisions. In addition, the evaluation of forecasts is done on the basis of testing the NAR block of historical data and on the basis of VAR block forecast in the form of mathematical expectation and standard deviation.

Nonmonotone line search methods with variable sample size

Nonmonotone line search methods for unconstrained minimization with the objective functions in the form of mathematical expectation are considered. The objective function is approximated by the sample average approximation (SAA) with a large sample of fixed size. The nonmonotone line search framework is embedded with a variable sample size strategy such that different sample size at each iteration allow us to reduce the cost of the sample average approximation. The variable sample scheme we consider takes into account the decrease in the approximate objective function and the quality of the approximation of the objective function at each iteration and thus the sample size may increase or decrease at each iteration. Nonmonotonicity of the line search combines well with the variable sample size scheme as it allows more freedom in choosing the search direction and the step size while the sample size is not the maximal one and increases the chances of finding a global solution. Eventually the maximal sample size is used so the variable sample size strategy generates the solution of the same quality as the SAA method but with significantly smaller number of function evaluations. Various nonmonotone strategies are compared on a set of test problems.

Line search methods with variable sample size for unconstrained optimization

Minimization of unconstrained objective functions in the form of mathematical expectation is considered. The Sample Average Approximation (SAA) method transforms the expectation objective function into a real-valued deterministic function using a large sample and thus deals with deterministic function minimization. The main drawback of this approach is its cost. A large sample of the random variable that defines the expectation must be taken in order to get a reasonably good approximation and thus the sample average approximation method requires a very large number of function evaluations. We present a line search strategy that uses variable sample size and thus makes the process significantly cheaper. Two measures of progress—lack of precision and a decrease of function value are calculated at each iteration. Based on these two measures a new sample size is determined. The rule we present allows us to increase or decrease the sample size at each iteration until we reach some neighborhood of the solution. An additional safeguard check is performed to avoid unproductive sample decrease. Eventually the maximal sample size is reached so that the variable sample size strategy generates a solution of the same quality as the SAA method but with a significantly smaller number of function evaluations. The algorithm is tested on a couple of examples, including the discrete choice problem.