Reducing the Dimensions of the Stochastic Programming Problems of Metallurgical Design Procedures

Process design procedures under uncertainty result in stochastic optimization problems whose resolution is complex due to the large uncertainty space, which hinders the application of optimization approaches, as well as the establishment of relationships between input and output variables. On the other hand, supervised machine learning (SML) offers tools with which to develop surrogate models, which are computationally inexpensive and efficient. This paper proposes a procedure based on modern design of experiments, deterministic optimization, SML tools, and global sensitivity analysis (GSA) to reduce the size of the uncertainty space for stochastic optimization problems. The proposal is illustrated with a case study based on the stochastic design of flotation plants. The results reveal that surrogate models of stochastic formulation enable the prediction of the structure, profitability parameters, and metallurgical parameters of designed flotation plants, as well as reducing the size of the uncertainty space via GSA and, consequently, establishing relationships between the input and output variables of the stochastic formulation.

Features of application of the Lanchester-type mathematical models in stochastic formulation when assessing the realities of air-land battle

This study provides a brief overview of the application of possible modifications of Lanchester-type models, namely, the representation of differential equations of such models in stochastic form. The stochastic setting of differential levels is used in Dynamic models if it is necessary to take into account the influence of random fluctuations (in particular, in radio engineering, thermodynamics, population dynamics models, etc.). As for Lanchester-type models, their stochastic appearance would allow considering the influence of random factors and elements of uncertainty, which are present to a certain extent in any combat operations. At the same time, unlike deterministic models, the numerical solution of systems of stochastic differential equations in such models requires the use of special methods, the choice of a specific one may be based on the requirements for the need to obtain an unambiguous approximate solution, or the probability distribution of the desired quantities. The possibility of obtaining different types of solutions is due to a characteristic feature of the developed methods for numerical integration of stochastic differential equations, namely, the existence of weak and strong approximate methods for solving them. For Lanchester equations, as models for predicting the probable course and results of combat operations, it seems appropriate to obtain a solution precisely in the form of parameters for distributions of random variables, which is possible after processing the results of using weak numerical methods. In addition, such methods are considered easier to implement in practice. Of particular note are the issues of estimating the stability of solutions (in the sense of Lyapunov) of stochastic models. While for Lanchester-type models, approximate practical methods for estimating stability can be considered, especially in relation to the simplest, linear statements of basic equations. The study considers an example of using the stochastic Lanchester-type model based on a system of linear inhomogeneous differential equations, with assumptions about the stability of solutions to the stochastic formulation of such equations.

Unsupervised Metric Learning with Synthetic Examples

Distance Metric Learning (DML) involves learning an embedding that brings similar examples closer while moving away dissimilar ones. Existing DML approaches make use of class labels to generate constraints for metric learning. In this paper, we address the less-studied problem of learning a metric in an unsupervised manner. We do not make use of class labels, but use unlabeled data to generate adversarial, synthetic constraints for learning a metric inducing embedding. Being a measure of uncertainty, we minimize the entropy of a conditional probability to learn the metric. Our stochastic formulation scales well to large datasets, and performs competitive to existing metric learning methods.