Faculty of Mechanical Engineering

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A Review of Modeling Approaches and Key Parameters in the Simulation of Wastewater Treatment Plants
(MDPI AG, 2026-01-20)
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SCALE INVARIANT STOCHASTIC GRADIENT METHOD WITH MOMENTUM
(Matematichki Bilten, Union of Mathematicians of Macedonia, 2023)
Nikolovski, Filip
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Optimization in noisy environments arises frequently in applications. Solving this problem quickly, efficiently, and accurately is therefore of great importance. The stochastic gradient descent (SGD) method has proven to be a fundamental and an effective tool which is flexible enough to allow modifications for improving its convergence properties. In this paper we propose a new algorithm for solving an unconstrained optimization problems in noisy environments which combines the SGD with a modified momentum term using a twopoint step size estimation in the Barzilai-Borwein (BB) framework. We perform a high probability analysis for the proposed algorithm and we establish its convergence under the standard assumptions. Numerical experiments demonstrate a promising behavior of the proposed method compared to the "vanilla" SGD with momentum in noise-free and in noisy environment when the objective function is scaled.
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ON THE CONVERGENCE OF THE PROXIMAL GRADIENT METHOD WITH VARIABLE STEP SIZES
(Union of Mathematicians of Macedonia, 2025)
Nikolovski, Filip
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Composite optimization problems arise frequently in modeling, since the objective function might contain components that do not possess some “nice” properties like differentiability; the case of l1 (LASSO) regularization is one such example. The proximal gradient methods are designed to handle this kind of optimization problems, and can solve them efficiently when the proximal mapping has a closed-form solution. Theoretical analyses of the convergence properties of the proximal gradient method with constant step size have showed sublinear and linear convergence for convex and strongly convex objective functions respectively. In this paper we show that under standard assumptions the same kind of convergence result can be established for the proximal gradient method with variable step sizes in the general setting of bounded step sizes. Further, a recently proposed step size selection for the proximal gradient method with variable step sizes is considered, and the above convergence analysis is implemented for this method.
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Gradient Descent Methods for Regularized Optimization
(Macedonian Academy of Sciences and Arts, 2024)
Nikolovski, Filip
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Regularization is a widely recognized technique in mathematical optimization. It can be used to smooth out objective functions, refine the feasible solution set, or prevent overfitting in machine learning models. Due to its simplicity and robustness, the gradient descent (GD) method is one of the primary methods used for numerical optimization of differentiable objective functions. However, GD is not well-suited for solving l1 regularized optimization problems since these problems are non-differentiable at zero, causing iteration updates to oscillate or fail to converge. Instead, a more effective version of GD, called the proximal gradient descent employs a technique known as soft-thresholding to shrink the iteration updates toward zero, thus enabling sparsity in the solution. Motivated by the widespread applications of proximal GD in sparse and low-rank recovery across various engineering disciplines, we provide an overview of the GD and proximal GD methods for solving regularized optimization problems. Furthermore, this paper proposes a novel algorithm for the proximal GD method that incorporates a variable step size. Unlike conventional proximal GD, which uses a fixed step size based on the global Lipschitz constant, our method estimates the Lipschitz constant locally at each iteration and uses its reciprocal as the step size. This eliminates the need for a global Lipschitz constant, which can be impractical to compute. Numerical experiments we performed on synthetic and real-data sets show notable performance improvement of the proposed method compared to the conventional proximal GD with constant step size, both in terms of number of iterations and in time requirements.
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Statistical Analysis of Urban Noise Measurements Data: Case Study for the City of Skopje
(EuroRegio, 2022)
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Nikolovski, Filip
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Noise from the road transport, particularly from vehicles in urban city areas largely accounts for the general noise level and annoyance of the citizens. The numerous volumes of motor vehicles flow can be treated statistically, which can establish a deeper insight into the contribution of the road noise to the prevalent noise pollution and its' characteristics. According to ISO 362 and ISO 1996:2, the environmental noise level from traffic is highly dependent on the vehicle category regarding the factor of contribution to the overall urban noise level. The purpose of this study is to analyze the dependence between the number and types of vehicles and measured standardized parameters (Leq, LAF and L95) for noise level assessment by implementing a statistical model analysis ofthe collected results. The number and the type of the vehicles is obtained from the States' traffic management and control center for a chosen road in the center of the city, whereas noise level measurements have been conducted with a Bruel&Kjaer sound level meter by using a standardized noise level measurement methodology procedure for the selected period on the given location. This study provides a detailed statistical approach of the collected noise and traffic volume data to obtain conclusions and prediction models for further management of the noise pollution problem in the city.
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Teaching probability using computer‐based simulations
(Faculty of Natural Sciences and Mathematics, 2022)
Nikolovski, Filip
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In this paper we develop and present a number of examples in which probabilities of events are modeled using computer‐based simulations. The principal goal of this approach is to help students who have started learning probability and statistics understand the concepts in those fields. In the authors' opinion, the outlined approach possesses several advantages, namely: it engages students in logical and critical thinking in its implementation, it offers great flexibility, adaptability, and possibility for generalization, it allows working with more advanced problems, and it leads to an easier acquisition of some notions and processes which are used in further application of probability theory and in statistics. The selected examples have been implemented using interactive notebooks in the programming language Python and are uploaded to a public access server for ease of access and sharing.
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New Derivative-free Nonmonotone Line Search Methods for Unconstrained Minimization
(2013)
Nikolovski, Filip
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Two new derivative-free nonmonotone line search methods for unconstrained optimization are proposed and analyzed. Convergence is established under standard conditions. Numerical results show good performance of the proposed methods.
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A Line Search Method with Memory for Unconstrained Optimization of Noisy Functions
(Union of Mathematicians of Macedonia, 2015)
Natasa Krejic
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Zorana Luzanin
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Nikolovski, Filip
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We propose a new line search method for unconstrained optimization of noisy functions. The nonmonotone line search rule is based on Ulbrich’s nonmonotone component [SIAM J. on Optimiz., 11 (4) (2001), 889–917]. The method uses only nosy functional values. Convergence under standard assumptions is established. Computational results show a good performance of the method compared with the monotone one.
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Complex-step derivative approximation in noisy environment
(Elsevier BV, 2018-01)
Nikolovski, Filip
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The complex-step derivative approximation is a powerful method for derivative approximations which has been successfully implemented in deterministic numerical algorithms. We explore and analyze its implementation in noisy environment through examples, error analysis and application to optimization methods. Numerical results show a promising performance of the complex-step gradient approximation in noisy environment.
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A nonmonotone line search method for noisy minimization
(Springer Science and Business Media LLC, 2015-01-24)
Krejić, Nataša
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Lužanin, Zorana
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Nikolovski, Filip
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A nonmonotone line search method for optimization in noisy environment is proposed. The method is defined for arbitrary search directions and uses only the noisy function values. Convergence of the proposed method is established under a set of standard assumptions. The computational issues are considered and the presented numerical results affirm that nonmonotone strategies are worth considering. Four different line search rules with three different directions are compared numerically. The influence of nonmonotonicity is discussed.

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