J. Brinkhuis (Jan)
http://repub.eur.nl/ppl/982/
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RePub, Erasmus University RepositoryConvex Duality and Calculus: Reduction to Cones
http://repub.eur.nl/pub/16362/
Sun, 01 Nov 2009 00:00:01 GMT<div>J. Brinkhuis</div>
An attempt is made to justify results from Convex Analysis by means of one method. Duality results, such as the Fenchel-Moreau theorem for convex functions, and formulas of convex calculus, such as the Moreau-Rockafellar formula for the subgradient of the sum of sublinear functions, are considered. All duality operators, all duality theorems, all standard binary operations, and all formulas of convex calculus are enumerated. The method consists of three automatic steps: first translation from the given setting to that of convex cones, then application of the standard operations and facts (the calculi) for convex cones, finally translation back to the original setting. The advantage is that the calculi are much simpler for convex cones than for other types of convex objects, such as convex sets, convex functions and sublinear functions. Exclusion of improper convex objects is not necessary, and recession directions are allowed as points of convex objects. The method can also be applied beyond the enumeration of the calculi.A linear programming proof of the second order conditions of nonlinear programming
http://repub.eur.nl/pub/13561/
Sun, 01 Feb 2009 00:00:01 GMT<div>J. Brinkhuis</div>
In this note we give a new, simple proof of the standard first and second order necessary conditions, under the Mangasarian–Fromovitz constraint qualification (MFCQ), for non-linear programming problems. We work under a mild constraint qualification, which is implied by MFCQ. This makes it possible to reduce the proof to the relatively easy case of inequality constraints only under MFCQ. This reduction makes use of relaxation of inequality constraints and it makes use of a penalty function. The new proof is based on the duality theorem for linear programming; the proofs in the literature are based on results of mathematical analysis. This paper completes the work in a recent note of Birbil et al. where a linear programming proof of the first order necessary conditions has been given, using relaxation of equality constraints.The Lagrange multiplier rule revisited
http://repub.eur.nl/pub/12016/
Thu, 03 Apr 2008 00:00:01 GMT<div>J. Brinkhuis</div><div>V. Protassov</div>
In this paper we give a short novel proof of the well-known Lagrange multiplier rule, discuss the sources of the power of this rule and consider several applications of this rule. The new proof does not use the implicit function theorem and combines the advantages of two of the most well-known proofs: it provides the useful geometric insight of the elimination approach based on differentiable curves and technically it is not more complicated than the simple penalty approach.
Then we emphasize that the power of the rule is the reversal of order of the natural tasks, elimination and differentiation. This turns the hardest task,
elimination, from a nonlinear problem into a linear one. This phenomenon is illustrated by several convincing examples of applications of the rule to various areas. Finally we give three hints on the use of the rule.Duality and calculi without exceptions for convex objects
http://repub.eur.nl/pub/11891/
Mon, 31 Mar 2008 00:00:01 GMT<div>J. Brinkhuis</div>
The aim of this paper is to make a contribution to the
investigation of the roots and essence of convex analysis, and to
the development of the duality formulas of convex calculus. This
is done by means of one single method: firstly conify, then
work with the calculus of convex cones, which consists of three
rules only, and finally deconify. This generates all
definitions of convex objects, duality operators, binary
operations and duality formulas, all without the usual need
to exclude degenerate situations. The duality operator for convex
function agrees with the usual one, the Legendre-Fenchel
transform, only for proper functions. It has the advantage over
the Legendre-Fenchel transform that the duality formula holds for
improper convex functions as well. This solves a well-known
problem, that has already been considered in Rockafellar's Convex
Analysis (R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970). The value of this result is that it leads
to the general validity of the formulas of Convex Analysis that
depend on the duality formula for convex functions. The approach
leads to the systematic inclusion into convex sets of recession
directions, and a similar extension for convex functions. The
method to construct binary operations given in (ibidem) is
formalized, and this leads to some new duality formulas. An
existence result for extended solutions of arbitrary convex
optimization problems is given. The idea of a similar extension of
the duality theory for optimization problems is given.On a conic approach to convex analysis.
http://repub.eur.nl/pub/11706/
Tue, 18 Mar 2008 00:00:01 GMT<div>J. Brinkhuis</div>
Abstract. The aim of this paper is to make an attempt to justify the main results from Convex Analysis by one elegant tool, the conification method, which consists of three steps: conify, work with convex cones, deconify. It is based on the fact that the standard operations and facts (`the calculi') are much simpler for special convex sets, convex cones. By considering suitable classes of convex cones, we get the standard operations and facts for all situations in the complete generality that is required. The main advantages of this conification method are that the standard operations---linear image, inverse linear image, closure, the duality operator, the binary operations and the inf-operator---are defined on all objects of each class of convex objects---convex sets, convex functions, convex cones and sublinear functions---and that moreover the standard facts---such as the duality theorem---hold for all closed convex objects. This requires that the analysis is carried out in the context of convex objects over cosmic space, the space that is obtained from ordinary space by adding a horizon, representing the directions of ordinary space.Descent: An optimization point of view on different fields
http://repub.eur.nl/pub/19260/
Thu, 16 Aug 2007 00:00:01 GMT<div>J. Brinkhuis</div>
The aim of this paper is to present a novel, transparent approach to a well-established field: the deep methods and applications of the complete analysis of continuous optimization problems. Standard descents give a unified approach to all standard necessary conditions, including the Lagrange multiplier rule, the Karush–Kuhn–Tucker conditions and the second order conditions. Nonstandard descents lead to new necessary conditions. These can be used to give surprising proofs of deep central results of fields that are generally viewed as distinct from optimization: the fundamental theorem of algebra, the maximum and the minimum principle of complex function theory, the separation theorems for convex sets, the orthogonal diagonalization of symmetric matrices and the implicit function theorem. These optimization proofs compare favorably with the usual proofs and are all based on the same strategy. This paper is addressed to all practitioners of optimization methods from many fields who are interested in fully understanding the foundations of these methods and of the central results above.Optimalisering in financiering, economie en wiskunde: welke toepassingen zijn overtuigend?
http://repub.eur.nl/pub/7023/
Mon, 07 Nov 2005 00:00:01 GMT<div>J. Brinkhuis</div>
In deze paper wordt de stelling onderbouwd dat er drie redenen zijn waarom een toepassing van optimaliseringsmethoden overtuigend is: `nut', `inzicht' en `diepte'. Ieder van de drie wordt geillustreerd met eenkarakteristiek voorbeeld: de prijsformule voor opties van Black en Scholes (`nut'), het werk van Kydland en Presscot (`inzicht') en een bewijs van de hoofdstelling van de algebra (`diepte').A simple view on convex analysis and its applications
http://repub.eur.nl/pub/7024/
Thu, 03 Nov 2005 00:00:01 GMT<div>J. Brinkhuis</div><div>V. Tikhomirov</div>
Our aim is to give a simple view on the basics and applications of convex analysis. The essential feature of this account is the systematic use of the possibility to associate to each convex object---such as a convex set, a convex function or a convex extremal problem--- a cone, without loss of information. The core of convex analysis is the possibility of the dual description of convex objects, geometrical and algebraical, based on the duality of vectorspaces; for each type of convex objects, this property is encoded in an operator of duality, and the name of the game is how to calculate these operators. The core of this paper is a unified presentation, for each type of convex objects, of the duality theorem and the complete list of calculus rules.
Now we enumerate the advantages of the `cone'-approach. It gives a unified and transparent view on the subject. The intricate rules of the convex calculus all flow naturally from one common source. We have included for each rule a precise description of the weakest convenient assumption under which it is valid. This appears to be useful for applications; however, these assumptioons are usually not given. We explain why certain convex objects have to be excluded in the definition of the operators of duality: the collections of associated cones of the target of an operator of duality need not be closed (here `closed' is meant in an algebraic sense). This makes clear that the remedy is to take the closure of the target. As a byproduct of the cone approach, we have found the solution of the open problem of how to use the polar operation to give a dual description of arbitrary convex sets.
The approach given can be extended to the infinite-dimensional case.Novel insights into the multiplier rule
http://repub.eur.nl/pub/7026/
Thu, 29 Sep 2005 00:00:01 GMT<div>J. Brinkhuis</div><div>V. Protassov</div>
We present the Lagrange multiplier rule, one of the basic optimization methods, in a new way. Novel features include:
• Explanation of the true source of the power of the rule: reversal of tasks, but not the use of multipliers.
• A natural proof based on a simple picture, but not the usual technical derivation from the implicit function theorem.
• A practical method to avoid the cumbersome second order conditions.
• Applications from various areas of mathematics, physics, economics.
• Some hnts on the use of the rule.Matrix convex functions with applications to weighted centers for semidefinite programming
http://repub.eur.nl/pub/7025/
Wed, 31 Aug 2005 00:00:01 GMT<div>J. Brinkhuis</div><div>Z-Q. Luo</div><div>S. Zhang</div>
In this paper, we develop various calculus rules for general smooth matrix-valued functions and for the class of matrix convex (or concave) functions first introduced by Loewner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function -log X to study a new notion of weighted convex centers for semidefinite programming (SDP) and show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the calculus rules for matrix convex functions can be used in the implementation of barrier methods for optimization problems involving nonlinear matrix functions.A comprehensive view on optimization: reasonable descent
http://repub.eur.nl/pub/6852/
Fri, 10 Jun 2005 00:00:01 GMT<div>J. Brinkhuis</div>
Reasonable descent is a novel, transparent approach to a well-established field: the deep methods and applications of the complete analysis of continuous optimization problems. Standard reasonable descents give a unified approach to all standard necessary conditions, including the Lagrange multiplier rule, the Karush-Kuhn-Tucker conditions and the second order conditions. Nonstandard reasonable descents lead to new necessary conditions. These can be used to give surprising proofs of deep central results outside optimization: the fundamental theorem of algebra, the maximum and the minimum principle of complex function theory, the separation theorems for convex sets, the orthogonal diagonalization of symmetric matrices and the implicit function theorem. These optimization proofs compare favorably with the usual proofs and are all based on the same strategy. This paper is addressed to all practitioners of optimization methods from many fields who are interested in fully understanding the foundations of these methods and of the central results above.On the universal method to solve extremal problems
http://repub.eur.nl/pub/1903/
Fri, 07 Jan 2005 00:00:01 GMT<div>J. Brinkhuis</div>
Some applications of the theory of extremal problems to mathematics and economics are made more accessible to non-experts.
1.The following fundamental results are known to all users of mathematical techniques, such as economist, econometricians, engineers and ecologists: the fundamental theorem of algebra, the Lagrange multiplier rule, the implicit function theorem, separation theorems for convex sets, orthogonal diagonalization of symmetric matrices. However, full explanations, including rigorous proofs, are only given in relatively advanced courses for mathematicians. Here, we offer short ans easy proofs. We show that akk these results can be reduced to the task os solving a suitable extremal problem. Then we solve each of the resulting problems by a universal strategy.
2. The following three practical results, each earning their discoverers the Nobel prize for Economics, are known to all economists and aonometricians: Nash bargaining, the formula of Black and Scholes for the price of options and the models of Prescott and Kydland on the value of commitment. However, the great value of such applications of the theory of extremal problems deserves to be more generally appreciated. The great impact of these results on real life examples is explained. This, rather than mathematical depth, is the correct criterion for assessing their value.A D-induced duality and its applications
http://repub.eur.nl/pub/1058/
Thu, 07 Aug 2003 00:00:01 GMT<div>J. Brinkhuis</div><div>S. Zhang</div>
This paper attempts to extend the notion of duality for convex cones, by basing it on a predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the inner-product is replaced by a pre-specified conic ordering, defined by a convex cone D, and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the D-induced duality in the paper. We further introduce the notion of D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced dual cones and are convenient to use for some practical applications. Properties of the extended duality, including the extended bi-polar theorem, are proven. Furthermore, attention is paid to the computation and approximation of the D-induced dual objects. We discuss, as examples, applications of the newly introduced D-induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization.A D-induced duality and its applications
http://repub.eur.nl/pub/547/
Wed, 02 Oct 2002 00:00:01 GMT<div>J. Brinkhuis</div><div>S. Zhang</div>
This paper attempts to extend the notion of duality for convex cones, by basing it on a
predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard
definition of dual cones, in the sense that the nonnegativity of the inner-product is replaced
by a pre-specified conic ordering, defined by a convex cone D, and the inner-product itself is
replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed
the D-induced duality in the paper. Basic properties of the extended duality, including the
extended bi-polar theorem, are proven. Examples are give to show the applications of the
new results.A structural version of the theorem of Hahn-Banach
http://repub.eur.nl/pub/1713/
Fri, 14 Dec 2001 00:00:01 GMT<div>J. Brinkhuis</div>
We consider one of the basic results of functional analysis, the classical theorem of Hahn-Banach. This theorem gives the existence of a continuous linear functional on a given normed vectorspace extending a given continuous linear functional on a subspace with the same norm. In this paper we generalize this existence theorem to a result on the structure of the set of all these extensions.On the Duality Theory of Convex Objects
http://repub.eur.nl/pub/6848/
Mon, 13 Aug 2001 00:00:01 GMT<div>J. Brinkhuis</div><div>V. Tikhomirov</div>
We consider the classical duality operators for convex objects such as the polar of a convex set containing the origin, the dual norm, the Fenchel-transform of a convex function and the conjugate of a convex cone. We give a new, sharper, unified treatment of the theory of these operators, deriving generalized theorems of Hahn-Banach, Fenchel-Moreau and Dubovitsky-Milyutin for the conjugate of convex cones in not necessarily finite dimensional vector spaces and hence for all the other duality operators of convex objects.On the complexity of the primal self-concordant barrier method
http://repub.eur.nl/pub/1666/
Sun, 31 Dec 2000 00:00:01 GMT<div>J. Brinkhuis</div>
there is no abstract of this reportOn the Principle of Fermat-Lagrange for Mixed Smooth-Convex Extremal Problems
http://repub.eur.nl/pub/7762/
Wed, 04 Mar 1998 00:00:01 GMT<div>J. Brinkhuis</div>
A necessary condition - the Principle of Fermat-Lagrange - is offered for mixed smooth-convex optimization problems. This generalizes and unifies most of the known necessary conditions for concrete finite and infinite dimensional optimization problems of interest. The new idea in comparison with the unified version of Tikhomirov and others ([I-T], [A-T-F] and [T]) is that a geometrical construction of the principle is given. In the present set-up constraints are not mentioned explicitly, the feasibility set is allowed to vary in a non-standard way and the objective function is also allowed to vary. An equivalent analytical formulation is given as well; we propose a new standard form for optimization problems which allows greater flexibility.On the Galois module structure over CM-fields
http://repub.eur.nl/pub/15896/
Tue, 01 Dec 1992 00:00:01 GMT<div>J. Brinkhuis</div>
In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extension N of a given CM-field K are invariant under the involution on the set of all realizations of Δ over K which is induced by complex conjugation on K and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number field K are completely characterized by ramification and Galois module structure.