Milan Hladik

Charles University in Prague | CUNI · Department of Applied Mathematics

In binary classification, kernel-free quadratic support vector machines are proposed to avoid difficulties such as finding appropriate kernel functions or tuning their hyper-parameters. Furthermore, Universum data points, which do not belong to any class, can be exploited to embed prior knowledge into the corresponding models to improve the general...

The support vector classification-regression machine for k-class classification (K-SVCR) is a novel multi-class classification approach based on the “1-versus-1-versus-rest” structure. In this work, we suggested an efficient model by proposing the p-norm \((0<p< 1)\) instead of the 2-norm for the regularization term in the objective function of K-S...

The separation of two polyhedra by a family of parallel hyperplanes is a well-known problem with important applications in operations research,statistics and functional analysis. In this paper, we introduce a new algorithm for constructing a family of parallel hyperplanes that separates two disjoint polyhedra given by a system of linear inequalitie...

Imbalanced datasets are prominent in real-world problems. In such problems, the data samples in one class are significantly higher than in the other classes, even though the other classes might be more important. The standard classification algorithms may classify all the data into the majority class, and this is a significant drawback of most stan...

The absolute value equations (AVE) problem is an algebraic problem of solving Ax+|x|=b. So far, most of the research focused on methods for solving AVEs, but we address the problem itself by analysing properties of AVE and the corresponding solution set. In particular, we investigate topological properties of the solution set, such as convexity, bo...

The support vector classification-regression machine for K-class classification (K-SVCR) is a novel multi-class classification method based on the “1-versus-1-versus-rest” structure. In this paper, we propose a least squares version of K-SVCR named LSK-SVCR. Similarly to the K-SVCR algorithm, this method assesses all the training data into a “1-ver...

Multi-task learning (MTL) has emerged as a promising topic of machine learning in recent years, aiming to enhance the performance of numerous related learning tasks by exploiting beneficial information. During the training phase, most of the existing multi-task learning models concentrate entirely on the target task data and ignore the non-target t...

Motivated by recently published papers, we discuss the problem of solving absolute value equations of the form Ax+|x|=b. Specifically, we show the equivalence of two sufficient conditions for the unsolvability of such equations which are based on linear programming. Furthermore, we prove that two generalizations of sufficient conditions for unique...

Optimization problems are often subject to various kinds of inexactness or inaccuracy of input data. Here, we consider multiobjective linear programming problems, in which two kinds of input entries have the form of interval data. First, we suppose that the objectives entries are interval values, and, second, we suppose that we have an interval est...

Parametric linear systems with interval coefficients arise in many practical applications in engineering and other related areas. In this paper, we derive a connection between parametric systems and interval linear programming, where interval parametric linear systems can be used to describe the weak optimal solution set. Then, we discuss the branc...

Universum data that do not belong to any class of a classification problem can be exploited to utilize prior knowledge to improve generalization performance. In this paper, we design a novel parametric ν-support vector machine with universum data (\( \mathfrak {U} \)Par-ν-SVM). Unlabeled samples can be integrated into supervised learning by means o...

Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discus...

The Universum provides prior knowledge about data in the mathematical problem to improve the generalization performance of the classifiers. Several works have shown that the Universum twin support vector machine (\( \mathfrak {U} \)-TSVM) is an efficient method for binary classification problems. In this paper, we improve the \( \mathfrak {U} \)-TS...

Universum twin support vector machine (\( \mathfrak {U} \)-TSVM) is an efficient method for binary classification problems . In this paper, we improve the \( \mathfrak {U} \)-TSVM algorithm and propose an improved Universum twin bounded support vector machine (named as IUTBSVM) . Indeed, by introducing a different Lagrangian function for the primal...

We consider the problem of maximization of a convex quadratic form on a convex polyhedral set, which is known to be NP-hard. In particular, we focus on upper bounds on the maximum value. We investigate utilization of different vector norms (estimating the Euclidean one) and different objective matrix factorizations. We arrive at some kind of dualit...

In this paper, we study the absolute value equation (AVE) Ax-b=|x|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax-b=|x|$$\end{document}. One effective approach to ha...

We consider the linear complementarity problem with uncertain data modeled by intervals, representing the range of possible values. Many properties of the linear complementarity problem (such as solvability, uniqueness, convexity, finite number of solutions etc.) are reflected by the properties of the constraint matrix. In order that the problem ha...

Interval linear programming provides a mathematical model for optimization problems affected by uncertainty, in which the uncertain data can be independently perturbed within the given lower and upper bounds. Many tasks in interval linear programming, such as describing the feasible set or computing the range of optimal values, can be solved by the...

We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously, we want this enclosure to be tight and cheap to compute; unfortunately, these two objectives are conflicting. The review of the available literature shows th...

We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices, resulting in the total time complexity O(n4), where n is...

We consider the horizontal linear complementarity problem and we assume that the input data have the form of intervals, representing the range of possible values. For the classical linear complementarity problem, there are known various matrix classes that identify interesting properties of the problem (such as solvability, uniqueness, convexity, f...

We study the optimum correction of infeasible systems of linear inequalities through making minimal changes in the coefficient matrix and the right-hand side vector by using the Frobenius norm. It leads to a special structured unconstrained nonlinear and nonconvex problem, which can be reformulated as a one-dimensional parametric minimization probl...

In binary classification, kernel-free linear or quadratic support vector machines are proposed to avoid dealing with difficulties such as finding appropriate kernel functions or tuning their hyper-parameters. Furthermore, Universum data points, which do not belong to any class, can be exploited to embed prior knowledge into the corresponding models...

To reliably model real robot characteristics, interval linear systems of equations allow to describe families of problems that consider sets of values. This allows to easily account for typical complexities such as sets of joint states and design parameter uncertainties. Inner approximations of the solutions to the interval linear systems can be us...

In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares i...

We consider the quadratic optimization problem \(\max _{x \in C}\ x^{\mathrm {T}}Q x + q^{\mathrm {T}}x\), where \(C\subseteq {\mathbb {R}}^n\) is a box and \(r:= {{\,\mathrm{rank}\,}}(Q)\) is assumed to be \({\mathcal {O}}(1)\) (i.e., fixed). We show that this case can be solved in polynomial time for an arbitrary Q and \(q\). The idea is based on...

We discuss some basic concepts and present a numerical procedure for finding the minimum-norm solution of convex quadratic programs (QPs) subject to linear equality and inequality constraints. Our approach is based on a theorem of alternatives and on a convenient characterization of the solution set of convex QPs. We show that this problem can be r...

The recent paper (DOI: 10.1007/s10898-017-0537-6) suggests various practical tests (sufficient conditions) for checking pseudoconvexity of a twice differentiable function on an interval domain. The tests were implemented using interval extensions of the gradient and the Hessian of the function considered. In this paper, we present an alternative ap...

We discuss some basic concepts and present a numerical procedure for finding the minimum-norm solution of convex quadratic programs (QPs) subject to linear equality and inequality constraints. Our approach is based on a theorem of alternatives and on a convenient characterization of the solution set of convex QPs. We show that this problem can be r...

We consider the problem of maximization of a quadratic form over a box. We identify the NP-hardness boundary for sparse quadratic forms: the problem is polynomially solvable for \(O(\log n)\) nonzero entries, but it is NP-hard if the number of nonzero entries is of the order \(n^{\varepsilon }\) for an arbitrarily small \(\varepsilon > 0\). Then we...

It is known that, in general, data processing under interval and fuzzy uncertainty is NP-hard—which means that, unless P = NP, no feasible algorithm is possible for computing the accuracy of the result of data processing. It is also known that the corresponding problem becomes feasible if the inputs do not interact with each other, i.e., if the dat...

Interval programming provides a mathematical model for uncertain optimization problems, in which the input data can be perturbed independently within the given lower and upper bounds. This paper discusses the recently proposed outcome range problem in the context of interval linear programming. The motivation for the outcome range problem is to ass...

We consider interval-valued pairwise comparison matrices and two types of consistency – weak (consistency for at least one realization) and strong (acceptable consistency for all realizations). Regarding weak consistency, we comment on the paper [Y. Dong and E. Herrera-Viedma, Consistency-Driven Automatic Methodology to Set Interval Numerical Scale...

Partially defined cooperative games are a generalisation of classical cooperative games in which payoffs for some of the coalitions are not known. In this paper we perform a systematic study of partially defined games, focusing on two important classes of cooperative games: convex games and positive games. In the first part, we focus on convexity a...

A rank estimator in robust regression is a minimizer of a function which depends (in addition to other factors) on the ordering of residuals but not on their values. Here we focus on the optimization aspects of rank estimators. We distinguish two classes of functions: a class with a continuous and convex objective function (CCC), which covers the c...

This paper deals with the fractional linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. A method is provided for the situation in which the feasible set is described by a linear interval system. Moreover, certain dependencies between the...

The support vector classification-regression machine for K-class classification (K-SVCR) is a novel multi-class classification method based on “1-versus-1-versus-rest” structure. In this paper, we propose a least squares version of K-SVCR named as LSK-SVCR. Similarly as the K-SVCR algorithm, this method assess all the training data into a “1-versus...

We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously we want this enclosure to be as tight as possible. The review of the available literature shows that in order to make a system more tractable most of the solu...

We deal with a system of parametric interval linear equations and also with its particular sub-classes defined by symmetry of the constraint matrix. We show that the problem of checking whether a given vector is a solution is a P-complete problem, meaning that there unlikely exists a polynomial closed form arithmetic formula describing the solution...

Recently, we have proposed several improvements of the standard monotonicity approach to solving parametric interval linear systems. The obtained results turned out to be very promising; i.e., we have achieved narrower bounds, while generally preserving the computational time. In this paper we propose another improvements, which aim to further decr...

It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some...

In this paper, we consider interval multi-objective linear programming (IMOLP) models which are used to deal with uncertainties of real-world problems. So far, a variety of approaches for obtaining efficient solutions (ESs) of these problems have been developed. In this paper, we propose a new and two generalized methods. In the new method, convert...

We consider a system of linear equations, whose coefficients depend linearly on interval parameters. Its solution set is defined as the set of all solutions of all admissible realizations of the parameters. We study unbounded directions of the solution set and its relation with its kernel. The kernel of a matrix characterizes unbounded direction in...

We consider the linear regression model with stochastic regressors and stochastic errors both in regressors and the dependent variable (“structural EIV model”), where the regressors and errors are assumed to satisfy some interesting and general conditions, different from traditional assumptions on EIV models (such as Deming regression). The most in...

This paper deals with the problem of linear programming with inexact data represented by real intervals. We introduce the concept of duality gap to interval linear programming. We give characterizations of strongly and weakly zero duality gap in interval linear programming and its special case where the matrix of coefficients is real. We show compu...

Maximization of a convex quadratic form on a convex polyhedral set is an NP-hard problem. We focus on computing an upper bound based on a factorization of the quadratic form matrix and employment of the maximum vector norm. Effectivity of this approach depends on the factorization used. We discuss several choices as well as iterative methods to imp...

We propose an exact iterative algorithm for minimization of a class of continuous cell-wise linear convex functions on a hyperplane arrangement. Our particular setup is motivated by evaluation of so-called rank estimators used in robust regression, where every cell of the underlying arrangement corresponds to a permutation of residuals (and we also...

Absolute value equations, due to their relation to the linear complementarity problem , have been intensively studied recently. In this paper, we present error bounds for absolute value equations. Along with the error bounds, we introduce an appropriate condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We d...

Interval programming provides one of the modern approaches to modeling optimization problems under uncertainty. Traditionally, the best and the worst optimal values determining the optimal value range are considered as the main solution concept for interval programs. In this paper, we present the concept of semi-strong values as a generalization of...

We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices. We present a method for general interval matrices as well...

We consider the quadratic optimization problem $\max_{x \in C}\ x^T Q x + q^T x$, where $C\subseteq\mathbb{R}^n$ is a box and $r := \mathrm{rank}(Q)$ is assumed to be $\mathcal{O}(1)$ (i.e., fixed). We show that this case can be solved in polynomial time for an arbitrary $Q$ and $q$. The idea is based on a reduction of the problem to enumeration of...

A rank estimator in robust regression is a minimizer of a function which depends (in addition to other factors) on the ordering of residuals but not on their values. Here we focus on the optimization aspects of rank estimators. We distinguish two classes of functions: the class with a continuous and convex objective function (CCC), which covers the...

Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb independently within the given lower and upper bounds. However, contrarily to classical linear programming, an interv...

We consider the linear complementarity problem with uncertain data modeled by intervals, representing the range of possible values. Many properties of the linear complementarity problem (such as solvability, uniqueness, convexity, finite number of solutions etc.) are reflected by the properties of the constraint matrix. In order that the problem ha...

In this paper, we investigate the parametric generalised Sylvester matrix equation A(p)XB(p)+C(p)XD(p)=F(p), whose elements are linear functions of uncertain parameters varying within intervals. This model generalises both interval matrix equations and parametric interval linear systems, so it is a quite general model. First, we give some sufficien...

We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running...

We propose a novel DEA method for computing efficiency scores. The method is based on a robust optimization viewpoint: the higher scores for those decision making units (DMU's) that remain efficient even for larger simultaneous and independent variations of all data and vice versa. Moreover, the value of each score itself gives the distance to inef...

Interval programming provides a mathematical tool for dealing with uncertainty in optimization problems. In this paper, we study two properties of interval linear programs: weak optimality and strong boundedness. The former property refers to the existence of a scenario possessing an optimal solution, or the problem of deciding non-emptiness of the...

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bou...

The aim of this paper is to obtain the range set for a given multiobjective linear programming problem and a weakly efficient solution. The range set is the set of all values of a parameter such that a given weakly efficient solution remains efficient when the objective coefficients vary in a given direction. The problem was originally formulated b...

In the first part of the paper, we consider standard systems of linear interval equations and we focus particularly on two solution methods, the Bauer–Skeel method and the Hansen–Bliek–Rohn method. We show relations between these two methods and between various modifications that are based on preconditioning of the system and on the residual form....

In the paper the interval least squares approach to estimate/fit data with interval uncertainties is introduced. The solution of this problem is discussed from the perspective of interval linear algebra. Using the interval linear algebra carefully, it is possible to significantly speed up the computation in specialized cases. The interval least squ...

The recent paper (DOI: 10.1007/s10898-017-0537-6) suggests various practical tests (sufficient conditions) for checking pseudoconvexity of a twice differentiable function on an interval domain. The tests were implemented using interval extensions of the gradient and the Hessian of the function considered. In this paper, we present an alternative ap...

This paper deals with interval parametric linear systems with general dependencies. Motivated by the so‐called parameterized solution introduced by Kolev, we consider the enclosures of the solution set in a revised affine form. This form is advantageous to a classical interval solution because it enables us to obtain both outer and inner bounds for...

Determining the set of all optimal solutions of a linear program with interval data is one of the most challenging problems discussed in interval optimization. In this paper, we study the topological and geometric properties of the optimal set and examine sufficient conditions for its closedness, boundedness, connectedness and convexity. We also pr...

We consider a linear programming problem, in which possibly all coefficients are subject to uncertainty in the form of deterministic intervals. The problem of computing the worst case optimal value has already been thoroughly investigated in the past. Notice that it might happen that the value can be infinite due to infeasibility of some instances....

The radius of regularity, sometimes spelled as the radius of nonsingularity, is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property...

In this paper we shed more light on determinants of real interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both relative and absolute approximation is proved. Next, methods computing verified enclosures of interval determin...

We investigate the shape of the optimal value of a linear programming problem with fuzzy-number coefficients. We build on the classical and also very recent results from interval linear programming as well as from parametric programming. We show that under general assumptions the optimal value is a well-defined fuzzy number. Its shape is piecewise...

In this paper we shed more light on determinants of interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both relative and absolute approximation is proved. Next, methods computing verified enclosures of interval determinants...

Background: Breath detection, i.e. its precise delineation in time is a crucial step in lung function data analysis as obtaining any clinically relevant index is based on the proper localization of breath ends. Current threshold or smoothing algorithms suffer from severe inaccuracy in cases of suboptimal data quality. Especially in infants, the pr...

We study the problem of checking pseudoconvexity of a twice differentiable function on an interval domain. Based on several characterizations of pseudoconvexity of a real function, we propose sufficient conditions for verifying pseudoconvexity on a domain formed by a Cartesian product of real intervals. We carried out numerical experiments to show...

The radius of regularity sometimes spelled as the radius of nonsingularity is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property i...

We implemented several known algorithms for finding an interval enclosure of the solution set of a linear system with linearly dependent interval parameters. To do that we have chosen MATLAB environment with use of INTLAB and VERSOFT libraries. Because our implementation is tested on Toeplitz and symmetric matrices, among others, there is a problem...

This paper deals with the problem of linear programming with inexact data represented by real closed intervals. Optimization problems with interval data arise in practical computations and they are of theoretical interest for more than forty years. We extend the concept of duality gap (DG), the difference between the primal and its dual optimal val...

In the recent years, there has been an intensive research of absolute value equations \(Ax-b=B|x|\). Various methods were developed, but less attention has been paid to approximating or bounding the solutions. We start filling this gap by proposing several outer approximations of the solution set. We present conditions for unsolvability and for exi...

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equival...

The presentation generalizes a known method for determining the regularity radius of an interval matrix ro the case of a parametric interval matrix.

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols , which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine...

It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some...

We investigate multiobjective linear programming with uncertain cost coefficients. We assume that lower and upper bounds for uncertain values are known, no other assumption on uncertain costs is needed. We focus on the so called possibly efficiency, which is defined as efficiency of at least one realization of interval coefficients. We show many fa...

This paper presents and describes in details an original method developed to solve over-constrained systems of non-linear interval equations that arise namely in parameter identification problems deriving from physical models and uncertain measurements. Our approach consists of computing an interval enclosure of the least square solution set and an...

We consider a linear programming problem with interval data. We discuss the problem of checking whether a given solution is optimal for each realization of interval data. This problem was studied for particular forms of linear programming problems. Herein, we extend the results to a general model and simplify the overall approach. Moreover, we insp...

We are concerned with the so called formal solution of an interval system of linear equations. We focus on the case where the coefficient matrix is deterministic (real) and the right-hand side is an interval vector. We show that the set of formal solutions represents a convex polyhedral set. We propose new properties of the formal solution related...

We introduce the tolerance approach to the construction of fuzzy regression coefficients of a possibilistic linear regression model with fuzzy data (both input and output). The method is very general: the only assumption is that α-cuts of the fuzzy data are efficiently computable. We take into account possible prior restrictions of the parameters s...

When we speak about parametric programming, sensitivity analysis, or related topics, we usually mean the problem of studying specified perturbations of the data such that for a given optimization problem some optimality criterion remains satisfied. In this paper, we turn to another question. Suppose that $A$ is a matrix having a specific property $...

We consider convex underestimators that are used in the global optimization αBB method and its variants. The method is based on augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symb...

This paper addresses the problem of computing the minimal and the maximal optimal value of a convex quadratic programming (CQP) problem when the coefficients are subject to perturbations in given intervals. Contrary to the previous results concerning on some special forms of CQP only, we present a unified method to deal with interval CQP problems....

Interval programming is a modern tool for dealing with uncertainty in practical optimization problems. In this paper, we consider a special class of interval linear programs with interval coefficients occurring only in the objective function and the right-hand-side vector, i.e. programs with a fixed (real) coefficient matrix. The main focus of the...

We propose a new approach to computing a parametric solution (the so-called p-solution) to parametric interval linear systems. Solving such system is an important part of many scientific and engineering problems involving uncertainties. The parametric solution has many useful properties. It permits to compute an outer solution, an inner estimate of...

Consider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by $\forall\exists$-quantification. Herein, we deal with the problem what properties must the coefficient matrix have in o...

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equival...