Econometric Institute Research Papers, Dec 3, 2015
In this note, a simplified version of the four main results for discrete-time infinite horizon pr... more In this note, a simplified version of the four main results for discrete-time infinite horizon problems, theorems 4.2-4.5 from Stokey, [SLP], is presented. A novel assumption on these problems is proposed-the uniform limit condition, which is formulated in terms of the data of the problem. It can be used for example before one has started to look for the optimal value function and for an optimal plan or if one cannot find them analytically: one verifies the uniform limit condition and then one disposes of criteria for optimality of the value function and a plan in terms of the functional equation and the boundedness condition. A comparison to [SLP] is made. The version in [SLP] requires one to verify whether a candidate optimal value function satisfies the boundedness condition; it is easier to check the uniform limit condition instead, as is demonstrated by examples. There is essentially no loss of strength or generality compared to [SLP]. The necessary and sufficient conditions for optimality coincide in the present paper but not in [SLP]. The proofs in the present paper are shorter than in [SLP]. An earlier attempt to simplify, in Acemoglu (2009)-here the limit condition is used rather than the uniform limit condition-is not correct.
• Why. Convex optimization includes linear programming, quadratic programming, semidefinite progr... more • Why. Convex optimization includes linear programming, quadratic programming, semidefinite programming, least squares problems and shortest distance problems. It is necessary to have theoretical tools to solve these problems. Finding optimal solutions exactly or by means of a law that characterizes them, is possible for a small minority of problems, but this minority contains very interesting problems. Therefore, most problems have to be solved numerically, by an algorithm. An analysis of the performance of algorithms for convex optimization requires techniques that are different from those presented in this book; therefore such an analysis falls outside the scope of this book.
Why. The aim of this chapter is to teach by means of some additional examples the craft of making... more Why. The aim of this chapter is to teach by means of some additional examples the craft of making a complete analysis of a convex optimization problem. These examples will illustrate all theoretical concepts and results in this book. This phenomenon is in the spirit of the quote by Cervantes. Enjoy watching the frying of eggs in this chapter and then fry some eggs yourself! • What. In this chapter, the following problems are solved completely; in brackets the technique that they illustrate is indicated. -Least squares (convex Fermat). -Generalized Fermat-Weber location problem (convex Fermat). -RAS method (KKT). -How to take a penalty (minimax and saddle point). -Ladies Diary problem (duality theory). -Second welfare theorem (KKT). -Minkowski's theorem on an enumeration of convex polytopes (KKT). -Duality for LP (duality theory). -Solving LP by taking a limit (the interior point algorithms are based on convex analysis).
• Why. The central object of convex analysis is a convex set. Whenever weighted averages play a r... more • Why. The central object of convex analysis is a convex set. Whenever weighted averages play a role, such as in the analysis by Nash of the question ‘what is a fair bargain?’, one is led to consider convex sets.
• Why. The phenomenon duality for a proper closed convex function, the possibility to describe it... more • Why. The phenomenon duality for a proper closed convex function, the possibility to describe it from the outside of its epigraph, by graphs of affine (constant plus linear) functions, has to be investigated. This has to be done for its own sake and as a preparation for the duality theory of convex optimization problems. An illustration of the power of duality is the following task, which is challenging without duality but easy if you use duality: prove the convexity of the main function from geometric programming, \(\ln (e^{x_1}+\cdots +e^{x_n})\), a smooth function that approximates the non-smooth function \(\max (x_1, \ldots , x_n)\).
In this note, a simplified version of the four main results for discrete-time infinite horizon pr... more In this note, a simplified version of the four main results for discrete-time infinite horizon problems, theorems 4.2-4.5 from Stokey, Lucas and Prescott (1989) [SLP], is presented. A novel assumption on these problems is proposed—the uniform limit condition, which is formulated in terms of the data of the problem. It can be used for example before one has started to look for the optimal value function and for an optimal plan or if one cannot find them analytically: one verifies the uniform limit condition and then one disposes of criteria for optimality of the value function and a plan in terms of the functional equation and the boundedness condition. A comparison to [SLP] is made. The version in [SLP] requires one to verify whether a candidate optimal value function satisfies the boundedness condition; it is easier to check the uniform limit condition instead, as is demonstrated by examples. There is essentially no loss of strength or generality compared to [SLP]. The necessary an...
textabstractA self-contained account of Morley's own proof of his celebrated trisector theore... more textabstractA self-contained account of Morley's own proof of his celebrated trisector theorem is given. This makes this elegant and almost forgotten fragment of analytic Euclidean geometry more accessible to modern readers
In this note, we consider a type of discrete-time infinite horizon problem that has one ingredien... more In this note, we consider a type of discrete-time infinite horizon problem that has one ingredient only, a constraint correspondence. The value function of a policy has an intuitive mono- tonicity property; this is the essence of the four standard theorems on the functional equation (‘the Bellman equation’). Some insight is offered into the boundedness condition for the value function that occurs in the formulation of these results: it can be interpreted as accountability of the loss of value caused by a non-optimal policy or, alternatively, it can be interpreted as irrelevance of devia- tions, in the distant future, from the considered policy. Without the boundedness condition, there is a gap, which can be viewed as the persistent potential positive impact of deviations, in the distant future, from the considered policy. The general stationary discrete-time infinite horizon optimization problem considered in Stokey and Lucas (1989) can be mapped to this type of problems and so the ...
This series contains compact volumes on the mathematical foundations of Operations Research, in p... more This series contains compact volumes on the mathematical foundations of Operations Research, in particular in the areas of continuous, discrete and stochastic optimization. Inspired by the PhD course program of the Dutch Network on the Mathematics of Operations Research (LNMB), and by similar initiatives in other territories, the volumes in this series offer an overview of mathematical methods for post-master students and researchers in Operations Research. Books in the series are based on the established theoretical foundations in the discipline, teach the needed practical techniques and provide illustrative examples and applications.
This paper is a contribution to the duality theory of extremal problems and prepares for a unifie... more This paper is a contribution to the duality theory of extremal problems and prepares for a unified approach to this theory and to Lagrange’s principle of elimination of constraints in the convex case, to be given in a forthcoming paper. We introduce additional – so-called vertical – solutions of dual extremal problems using a compactification construction. The new problem – called the complete dual problem – is solvable if it has a finite value. We introduce a set which is relatively easy to handle, the set of promising vertical slopes. This set contains the set of vertical solutions; moreover these two sets are equal if the complete dual problem is solvable. Finally, we give a geometric interpretation of the notions vertical solutions and promising vertical slopes in terms of pointed halfspaces.
Abstract. The aim of this paper is to make an attempt to justify the main results from Convex Ana... more 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 t...
We consider one of the basic results of functional analysis, the classical theorem of Hahn-Banach... more 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.
Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 1972
Dedicated to Professor J. Th. G. Overbeek on the occasion of the 25th anniversary of his appointm... more Dedicated to Professor J. Th. G. Overbeek on the occasion of the 25th anniversary of his appointment as a Professor of Physical Chemistry. * This only applies if the vacuum chamber does not move relative to the optical axis during evacuation. If it does, the light beam should be inspected only after the flat transparent disks have.been inserted in the lens holders and the chamber has been evacuated.
• A vector-space calculation of the derivative of the determinant function. • All conventions of ... more • A vector-space calculation of the derivative of the determinant function. • All conventions of matrix calculus arise naturally in the vector-space approach. • The vector-space approach clarifies the role of the Kronecker product in matrix calculus. • The vector-space approach gives a quick access to the v-space in interior point methods.
In this paper, we develop various calculus rules for general smooth matrix-valued functions and f... more 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 Löwner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function -log X to study a new notion of weighted 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.
Econometric Institute Research Papers, Dec 3, 2015
In this note, a simplified version of the four main results for discrete-time infinite horizon pr... more In this note, a simplified version of the four main results for discrete-time infinite horizon problems, theorems 4.2-4.5 from Stokey, [SLP], is presented. A novel assumption on these problems is proposed-the uniform limit condition, which is formulated in terms of the data of the problem. It can be used for example before one has started to look for the optimal value function and for an optimal plan or if one cannot find them analytically: one verifies the uniform limit condition and then one disposes of criteria for optimality of the value function and a plan in terms of the functional equation and the boundedness condition. A comparison to [SLP] is made. The version in [SLP] requires one to verify whether a candidate optimal value function satisfies the boundedness condition; it is easier to check the uniform limit condition instead, as is demonstrated by examples. There is essentially no loss of strength or generality compared to [SLP]. The necessary and sufficient conditions for optimality coincide in the present paper but not in [SLP]. The proofs in the present paper are shorter than in [SLP]. An earlier attempt to simplify, in Acemoglu (2009)-here the limit condition is used rather than the uniform limit condition-is not correct.
• Why. Convex optimization includes linear programming, quadratic programming, semidefinite progr... more • Why. Convex optimization includes linear programming, quadratic programming, semidefinite programming, least squares problems and shortest distance problems. It is necessary to have theoretical tools to solve these problems. Finding optimal solutions exactly or by means of a law that characterizes them, is possible for a small minority of problems, but this minority contains very interesting problems. Therefore, most problems have to be solved numerically, by an algorithm. An analysis of the performance of algorithms for convex optimization requires techniques that are different from those presented in this book; therefore such an analysis falls outside the scope of this book.
Why. The aim of this chapter is to teach by means of some additional examples the craft of making... more Why. The aim of this chapter is to teach by means of some additional examples the craft of making a complete analysis of a convex optimization problem. These examples will illustrate all theoretical concepts and results in this book. This phenomenon is in the spirit of the quote by Cervantes. Enjoy watching the frying of eggs in this chapter and then fry some eggs yourself! • What. In this chapter, the following problems are solved completely; in brackets the technique that they illustrate is indicated. -Least squares (convex Fermat). -Generalized Fermat-Weber location problem (convex Fermat). -RAS method (KKT). -How to take a penalty (minimax and saddle point). -Ladies Diary problem (duality theory). -Second welfare theorem (KKT). -Minkowski's theorem on an enumeration of convex polytopes (KKT). -Duality for LP (duality theory). -Solving LP by taking a limit (the interior point algorithms are based on convex analysis).
• Why. The central object of convex analysis is a convex set. Whenever weighted averages play a r... more • Why. The central object of convex analysis is a convex set. Whenever weighted averages play a role, such as in the analysis by Nash of the question ‘what is a fair bargain?’, one is led to consider convex sets.
• Why. The phenomenon duality for a proper closed convex function, the possibility to describe it... more • Why. The phenomenon duality for a proper closed convex function, the possibility to describe it from the outside of its epigraph, by graphs of affine (constant plus linear) functions, has to be investigated. This has to be done for its own sake and as a preparation for the duality theory of convex optimization problems. An illustration of the power of duality is the following task, which is challenging without duality but easy if you use duality: prove the convexity of the main function from geometric programming, \(\ln (e^{x_1}+\cdots +e^{x_n})\), a smooth function that approximates the non-smooth function \(\max (x_1, \ldots , x_n)\).
In this note, a simplified version of the four main results for discrete-time infinite horizon pr... more In this note, a simplified version of the four main results for discrete-time infinite horizon problems, theorems 4.2-4.5 from Stokey, Lucas and Prescott (1989) [SLP], is presented. A novel assumption on these problems is proposed—the uniform limit condition, which is formulated in terms of the data of the problem. It can be used for example before one has started to look for the optimal value function and for an optimal plan or if one cannot find them analytically: one verifies the uniform limit condition and then one disposes of criteria for optimality of the value function and a plan in terms of the functional equation and the boundedness condition. A comparison to [SLP] is made. The version in [SLP] requires one to verify whether a candidate optimal value function satisfies the boundedness condition; it is easier to check the uniform limit condition instead, as is demonstrated by examples. There is essentially no loss of strength or generality compared to [SLP]. The necessary an...
textabstractA self-contained account of Morley's own proof of his celebrated trisector theore... more textabstractA self-contained account of Morley's own proof of his celebrated trisector theorem is given. This makes this elegant and almost forgotten fragment of analytic Euclidean geometry more accessible to modern readers
In this note, we consider a type of discrete-time infinite horizon problem that has one ingredien... more In this note, we consider a type of discrete-time infinite horizon problem that has one ingredient only, a constraint correspondence. The value function of a policy has an intuitive mono- tonicity property; this is the essence of the four standard theorems on the functional equation (‘the Bellman equation’). Some insight is offered into the boundedness condition for the value function that occurs in the formulation of these results: it can be interpreted as accountability of the loss of value caused by a non-optimal policy or, alternatively, it can be interpreted as irrelevance of devia- tions, in the distant future, from the considered policy. Without the boundedness condition, there is a gap, which can be viewed as the persistent potential positive impact of deviations, in the distant future, from the considered policy. The general stationary discrete-time infinite horizon optimization problem considered in Stokey and Lucas (1989) can be mapped to this type of problems and so the ...
This series contains compact volumes on the mathematical foundations of Operations Research, in p... more This series contains compact volumes on the mathematical foundations of Operations Research, in particular in the areas of continuous, discrete and stochastic optimization. Inspired by the PhD course program of the Dutch Network on the Mathematics of Operations Research (LNMB), and by similar initiatives in other territories, the volumes in this series offer an overview of mathematical methods for post-master students and researchers in Operations Research. Books in the series are based on the established theoretical foundations in the discipline, teach the needed practical techniques and provide illustrative examples and applications.
This paper is a contribution to the duality theory of extremal problems and prepares for a unifie... more This paper is a contribution to the duality theory of extremal problems and prepares for a unified approach to this theory and to Lagrange’s principle of elimination of constraints in the convex case, to be given in a forthcoming paper. We introduce additional – so-called vertical – solutions of dual extremal problems using a compactification construction. The new problem – called the complete dual problem – is solvable if it has a finite value. We introduce a set which is relatively easy to handle, the set of promising vertical slopes. This set contains the set of vertical solutions; moreover these two sets are equal if the complete dual problem is solvable. Finally, we give a geometric interpretation of the notions vertical solutions and promising vertical slopes in terms of pointed halfspaces.
Abstract. The aim of this paper is to make an attempt to justify the main results from Convex Ana... more 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 t...
We consider one of the basic results of functional analysis, the classical theorem of Hahn-Banach... more 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.
Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, 1972
Dedicated to Professor J. Th. G. Overbeek on the occasion of the 25th anniversary of his appointm... more Dedicated to Professor J. Th. G. Overbeek on the occasion of the 25th anniversary of his appointment as a Professor of Physical Chemistry. * This only applies if the vacuum chamber does not move relative to the optical axis during evacuation. If it does, the light beam should be inspected only after the flat transparent disks have.been inserted in the lens holders and the chamber has been evacuated.
• A vector-space calculation of the derivative of the determinant function. • All conventions of ... more • A vector-space calculation of the derivative of the determinant function. • All conventions of matrix calculus arise naturally in the vector-space approach. • The vector-space approach clarifies the role of the Kronecker product in matrix calculus. • The vector-space approach gives a quick access to the v-space in interior point methods.
In this paper, we develop various calculus rules for general smooth matrix-valued functions and f... more 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 Löwner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function -log X to study a new notion of weighted 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.
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Papers by Jan Brinkhuis