arXiv subjects

Marko M. Mäkelä

Publications and source records attributed to Marko M. Mäkelä.

3 recordsLinked to original sources

arXiv2026

Inexact Limited Memory Bundle Method

Large-scale nonsmooth optimization problems arise in many real-world applications, but obtaining exact function and subgradient values for these problems may be computationally expensive or even infeasible. In many practical settings, only inexact information is available due to measurement or modeling errors, privacy-preserving computations, or stochastic approximations, making inexact optimization methods particularly relevant. In this paper, we propose a novel inexact limited memory bundle method for large-scale nonsmooth nonconvex optimization. The method tolerates noise in both function values and subgradients. We prove the global convergence of the proposed method to an approximate stationary point. Numerical experiments with different levels of noise in function and/or subgradient values show that the method performs well with both exact and noisy data. In particular, the results demonstrate competitiveness in large-scale nonsmooth optimization and highlight the suitability of the method for applications where noise is unavoidable, such as differential privacy in machine learning.

math.OC↗

arXiv2026

Clust-Splitter - an Efficient Nonsmooth Optimization-Based Algorithm for Clustering Large Datasets

Clustering is a fundamental task in data mining and machine learning, particularly for analyzing large-scale data. In this paper, we introduce Clust-Splitter, an efficient algorithm based on nonsmooth optimization, designed to solve the minimum sum-of-squares clustering problem in very large datasets. The clustering task is approached through a sequence of three nonsmooth optimization problems: two auxiliary problems used to generate suitable starting points, followed by a main clustering formulation. To solve these problems effectively, the limited memory bundle method is combined with an incremental approach to develop the Clust-Splitter algorithm. We evaluate Clust-Splitter on real-world datasets characterized by both a large number of attributes and a large number of data points and compare its performance with several state-of-the-art large-scale clustering algorithms. Experimental results demonstrate the efficiency of the proposed method for clustering very large datasets, as well as the high quality of its solutions, which are on par with those of the best existing methods.

cs.LG↗

arXiv2026

KKT conditions for optimization with generalized invex fuzzy functions

This paper explores optimality conditions in optimization problems involving generalized invex fuzzy functions. We extend the classical KKT framework to settings in which the objective and constraint functions are nonsmooth, vector-valued, and fuzzy-valued, and satisfy various generalized invexity conditions such as V-invexity, V-pseudoinvexity, and Vquasiinvexity. After reviewing key concepts from nonsmooth analysis and multiobjective optimization, we derive new KKT-type conditions under weaker assumptions than classical convexity, ensuring (weak) Pareto optimality in fuzzy environments. Our results unify and generalize earlier work by Antczak and Mishra as well as demonstrate the power of generalized invexity in establishing optimality without requiring differentiability nor convexity. Several illustrative examples are included to demonstrate the applicability of the developed theory.

math.OC↗