Skills (12)
Research experience
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Teaching: Game theory.
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Teaching: Functional analysis
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Teaching: Linear programming
Education
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Oct 1998–
Jun 2006Charles University
Operational Research · Ph.D.Czech Republic · Prague -
Oct 1993–
Sep 1998Charles University
Computer Science · M.Sc.Czech Republic · Prague
Other
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LanguagesEnglish, Russian, Czech.
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Scientific MembershipsCzech Society for Operations Research
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Other InterestsWalking, hiking, roving in the countryside. Philosophy., James Redfield:
--- The Celestine Prophecy.
--- The Tenth Insight: Holding the Vision.
--- The Secret of Shambhala: In Search of the Eleventh Insight. ...
Eckhart Tolle:
--- The Power of Now.
Questions and Answers (6) View all
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Answer added in Operational Research4 software to visualize or compute problems of LP, ORTo visualize… If it means to visualize the solution to an LP programme: I recall that they have been developing some tutorial software for students ... [more]
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Answer added in Operational Research4 software to visualize or compute problems of LP, ORTo visualize OR to compute? There are many pieces of software to compute (i.e. solve) problems of LP (linear programming). For example: GLPK http://... [more]
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Answer added in Mathematics14 The chain RuleSorry. The original answer in this comment was to the above comment. Nonetheless, the above comment was completely edited. Therefore, this answer h... [more]
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Answer added in Mathematics14 The chain Rule> If we have: z = f (x,y) and z = f (t), The function "f" is the same in both cases? Sincerely, David Bartl
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Answer added in Mathematics19 System of equationsJemimah, it is difficult to answer your question (whether you have to choose a norm if you use some maths software). It depends. Perhaps you can u... [more]
Publications (11) View all
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Article: On Fredholm's Theorem of the Alternative and a Corollary of Rohn's Residual Existence Theorem
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ABSTRACT: By Fredholm's Theorem of the alternative, the system ${\mib A} {\mib x} = {\mib b}$ of linear equations has no solution if and only if ${\mib u}^{\sl T} \! {\mib A} = {\mib o}^{\sl T}$ and ${\mib u}^{\sl T} {\mib b} \neq 0$ for some ${\mib u} \in {\msbm R}^m$. Recently, Rohn proved as a corollary of the Residual Existence Theorem for linear equations [{\it Optim.\ Lett.\/}\ 4 (2010), 287--292] that the system ${\mib A} {\mib x} = {\mib b}$ has a solution if and only if the residual set $\{\, {\mib A} {\mib x} - {\mib b} : {\mib x} \in {\msbm R}^n \,\}$ intersects all the orthants of~${\msbm R}^m$. We study the relation between both the results in the more general setting of a vector space over a linearly ordered (possibly skew) field, obtain a new proof of the corollary, and give a generalisation of Fredholm's Theorem of the alternative.Mathematica Pannonica. 12/2012; 23(2):311–320. -
Article: Farkas' Lemma, Gale's Theorem, and Linear Programming: the Infinite Case in an Algebraic Way
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ABSTRACT: We study a problem of linear programming in the setting of a vector space over a linearly ordered (possibly skew) field. The dimension of the space may be infinite. The objective function is a linear mapping into another linearly ordered vector space over the same field. In that algebraic setting, we recall known results: Farkas' Lemma, Gale's Theorem of the alternative, and the Duality Theorem for linear programming with finite number of linear constraints. Given that ``semi-infinite'' case, i.e.\ results for finite systems of linear inequalities in an infinite-dimensional space, we are motivated to consider the infinite case: infinite systems of linear inequalities in an infinite-dimensional space. Given such a system, we assume that only a finite number of the left-hand sides is non-zero at a point. We shall also assume a certain constraint qualification (CQ), presenting counterexamples violating the (CQ). Then, in the described setting, we formulate an infinite variant of Farkas' Lemma along with an infinite variant of Gale's Theorem of the alternative. Finally, we formulate the problem of an infinite linear programming, its dual problem, and the Duality Theorem for the problems.Global Journal of Mathematical Sciences (GJMS). 12/2012; 1(1):18–23. -
Conference Proceeding: Application of Cooperative Game Solution Concepts to a Collusive Oligopoly Game
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ABSTRACT: An oligopoly is a market where a couple of large producers supply some goods. If the oligopoly is collusive, the producers form coalitions. Then, within each of the coalitions, the producers wish to divide their total profit among themselves. They could use a cooperative transferable utility game solution concept, such as the core, if the game were in the coalitional form. This, however, is not the case. In this paper, we propose an approach to overcome that difficulty: converting the collusive oligopoly into the partition function form, we show how the known cooperative game solution concepts (core, bargaining set) can be applied to that game. Actually, the proposed approach is suitable not only for a collusive oligopoly, but, under some assumptions, for any cooperative strategic form game.MME 2012; 09/2012 -
Article: A note on the short algebraic proof of Farkas' Lemma
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ABSTRACT: The author published ``A Short Algebraic Proof of the Farkas Lemma'' [SIAM J. Optim.~19 (2008), pp.\ 234--239]. The author then found, to his opinion, a better way of the exposition of the proof. He would like therefore to publish the new form of the proof in this note.Linear and Multilinear Algebra 08/2012; 60(8):897–901. · 0.73 Impact Factor -
Article: Separation Theorems for Convex Polytopes and Finitely-Generated Cones Derived from Theorems of the Alternative
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ABSTRACT: We derive from Motzkin's Theorem that a point can be strongly separated by a hyperplane from a convex polytope and a finitely-generated convex cone. We state a similar result for Tucker's Theorem of the alternative. A generalisation of the residual existence theorem for linear equations which has recently been proved by Rohn [Optim.\ Lett.\ 4 (2010) 287--292] is a corollary. We state all the results in the setting of a general vector space over a linearly ordered (possibly skew) field.Linear Algebra and its Applications 05/2012; 436(9):3784–3789. · 0.97 Impact Factor
