C. Roos | TU Delft (original) (raw)

Papers by C. Roos

Reports of the Faculty of Technical Mathematics and …, 1998

Page 1. DELFT UNIVERSITY OF TECHNOLOGY REPORT 98{24 An Easy Way to Teach Interior Point Methods T... more Page 1. DELFT UNIVERSITY OF TECHNOLOGY REPORT 98{24 An Easy Way to Teach Interior Point Methods Tam as Terlaky ISSN 0922{5641 Reports of the Faculty of Technical Mathematics and Informatics 98{24 Delft June, 1998 Page 2. ...

... Theory and algorithms for linear optimization: An interior point approach. Post a Comment. CO... more ... Theory and algorithms for linear optimization: An interior point approach. Post a Comment. CONTRIBUTORS: Author: Roos, C. (b. 1941, d. ----. Author: Terlaky, T. Author: Vial, JP. PUBLISHER: Wiley (Chichester and New York). SERIES TITLE: YEAR: 1997. ...

Annals of Operations Research, 1995

The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly nonsmooth, ... more The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly nonsmooth, semi{in nite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein solve a linear approximation (localization) of the problem, and then generate an additional cut to remove the linear program's optimal point. Other methods like the \central cutting" plane methods of Elzinga{Moore and Go n{Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. Contrary to these existing techniques, we develop a method which does not solve the linear relaxations till optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the e ect of adding and deleting constraints in long{step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the classes of convex geometric and semi{in nite programming problems.

Computers & Mathematics with Applications, 2003

Citeseer

CiteSeerX - Document Details (Isaac Councill, Lee Giles): tions in this window will be described ... more CiteSeerX - Document Details (Isaac Councill, Lee Giles): tions in this window will be described in turn. Figure 1: The window in X-GAmeter. - The following functions are available from the option: : opens up a window from which user may load a data file. : saves the best solution ...

A standard quadratic problem consists of finding global maximizers of a quadratic form over the s... more A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization F F T where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

We present a polynomial-time primal-dual interior-point algorithm for solving linear optimization... more We present a polynomial-time primal-dual interior-point algorithm for solving linear optimization (LO) problems, based on generalized logarithmic barrier function. The growth term depends on a parameter p ∈ [0, 1]. The kernel functions are neither self-regular nor strongly convex. The classical logarithmic barrier function occurs if p = 1. The goal of this paper is to investigate such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. In order to achieve these complexity results, several new techniques had to be used for the analysis. Complexity issues are discussed and they are O n log n , and O √ n log n , for large-update and small-update methods, respectively. Numerical tests show that the iteration bounds are influenced by p. We conclude that a gap still exists between the theoretical complexity and practical behavior of the algorithm.

Operations Research Proceedings, 2005

ABSTRACT We investigate the robust shortest path problem using the robust linear optimization met... more ABSTRACT We investigate the robust shortest path problem using the robust linear optimization methodology as proposed by Ben-Tal and Nemirovski. We discuss two types of uncertainty, namely, box uncertainty and ellipsoidal uncertainty. In case of box uncertainty, the robust counterpart is simple. It is a shortest path problem with the original arc lengths replaced by their upper bounds. When dealing with ellipsoidal uncertainty, we obtain a conic quadratic optimization problem with binary variables. We present an example to show that a subpath of a robust shortest path is not necessarily a robust shortest path.

SIAM Journal on Optimization, 2002

We introduce a new barrier function which is not a barrier function in the usual sense: it has fi... more We introduce a new barrier function which is not a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, its iteration bound, O √ n log n log n ε ,i s as good as it can be: it is the best known bound for large-update methods. The recently introduced

SIAM Journal on Optimization, 2004

Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) ... more Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on these kernel functions. In both the algorithm and its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of kernel function in the new class. We apply this scheme to seven specific kernel functions. Some of these functions are self-regular, and others are not. One of the functions differs from the others, and from all self-regular functions, in the sense that its growth term is linear. Iteration bounds for both large-and small-update methods are derived. It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely, O( √ n log n ε ). For large-update methods the best obtained bound is O( √ n(log n) log n ε ), which until now has been the best known bound for such methods.

RAIRO - Operations Research, 2008

ABSTRACT In this paper we present a generic primal-dual interior point methods (IPMs) for linear ... more ABSTRACT In this paper we present a generic primal-dual interior point methods (IPMs) for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. The proposed kernel function does not satisfy all the conditions proposed in [2]. We show that the corresponding large-update algorithm improves the iteration complexity with a factor n 1 6 when compared with the method based on the use of the classical logarithmic barrier function. For small-update interior point methods the iteration bound is O ( √ n log n), which is currently the best-known bound for primal-dual ɛ IPMs.

IEEE Transactions on Information Theory, 1981

A well-known theorem of MacWilliams states that b is the distribution of the dual code Cl wheneve... more A well-known theorem of MacWilliams states that b is the distribution of the dual code Cl whenever the code C is line?.

We propose a strategy for building up the linear program while using a logarithmic barrier method... more We propose a strategy for building up the linear program while using a logarithmic barrier method.The method starts with a (small) subset of the dual constraints, and follows the correspondingcentral path until the iterate is close to (or violates) one of the constraints, which is in turn addedto the current system. This process is repeated until an optimal solution is

The frequency assignment problem FAP is the problem of assigning frequencies to transmission link... more The frequency assignment problem FAP is the problem of assigning frequencies to transmission links such that no interference between signals occurs. This implies distance constraints between assigned frequencies of links. The objective is to minimize the number of used frequencies. We present a n i n teger linear programming formulation that is closely related to the vertex packing problem. Although the size of this formulation is an order of magnitude larger than the underlying network of links, we use the integer linear programming formulation within a branch-and-cut algorithm. This algorithm employs problem speci c and generic techniques such as reduction methods, primal heuristics, and branching rules to obtain optimal solutions. We report on computational experience with real-life instances.

SIAM Journal on Optimization, 1997

In this paper the new polynomial affine scaling algorithm of Jansen, Roos andTerlaky for LP is ex... more In this paper the new polynomial affine scaling algorithm of Jansen, Roos andTerlaky for LP is extended to PSD linear complementarity problems. The algorithmis immediately further generalized to allow higher order scaling. These algorithmsare also new for the LP case. The analysis is based on Ling's proof for the LP case,hence allows an arbitrary interior feasible pair to start with.

Optimization, 1994

The literature in the field of interior point methods for linear programming has been almost excl... more The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Güler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple approach which is based on an embedding of the primal problem and its dual

Operations Research Letters, 1997

The formulation of interior point algorithms for semide nite programming has become an active res... more The formulation of interior point algorithms for semide nite programming has become an active research area, following the success of the methods for large{ scale linear programming. Many interior point methods for linear programming have now been extended to the more general semide nite case, but the initialization problem remained unsolved.

Mathematical Programming, 1997

In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming ... more In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen, Roos and Terlaky enhances an asymptotical O(f~n L) complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semi-definite linear complementarity problems.

Reports of the Faculty of Technical Mathematics and …, 1998

Page 1. DELFT UNIVERSITY OF TECHNOLOGY REPORT 98{24 An Easy Way to Teach Interior Point Methods T... more Page 1. DELFT UNIVERSITY OF TECHNOLOGY REPORT 98{24 An Easy Way to Teach Interior Point Methods Tam as Terlaky ISSN 0922{5641 Reports of the Faculty of Technical Mathematics and Informatics 98{24 Delft June, 1998 Page 2. ...

... Theory and algorithms for linear optimization: An interior point approach. Post a Comment. CO... more ... Theory and algorithms for linear optimization: An interior point approach. Post a Comment. CONTRIBUTORS: Author: Roos, C. (b. 1941, d. ----. Author: Terlaky, T. Author: Vial, JP. PUBLISHER: Wiley (Chichester and New York). SERIES TITLE: YEAR: 1997. ...

Annals of Operations Research, 1995

The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly nonsmooth, ... more The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly nonsmooth, semi{in nite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein solve a linear approximation (localization) of the problem, and then generate an additional cut to remove the linear program's optimal point. Other methods like the \central cutting" plane methods of Elzinga{Moore and Go n{Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. Contrary to these existing techniques, we develop a method which does not solve the linear relaxations till optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the e ect of adding and deleting constraints in long{step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the classes of convex geometric and semi{in nite programming problems.

Computers & Mathematics with Applications, 2003

Citeseer

CiteSeerX - Document Details (Isaac Councill, Lee Giles): tions in this window will be described ... more CiteSeerX - Document Details (Isaac Councill, Lee Giles): tions in this window will be described in turn. Figure 1: The window in X-GAmeter. - The following functions are available from the option: : opens up a window from which user may load a data file. : saves the best solution ...

A standard quadratic problem consists of finding global maximizers of a quadratic form over the s... more A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization F F T where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

We present a polynomial-time primal-dual interior-point algorithm for solving linear optimization... more We present a polynomial-time primal-dual interior-point algorithm for solving linear optimization (LO) problems, based on generalized logarithmic barrier function. The growth term depends on a parameter p ∈ [0, 1]. The kernel functions are neither self-regular nor strongly convex. The classical logarithmic barrier function occurs if p = 1. The goal of this paper is to investigate such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. In order to achieve these complexity results, several new techniques had to be used for the analysis. Complexity issues are discussed and they are O n log n , and O √ n log n , for large-update and small-update methods, respectively. Numerical tests show that the iteration bounds are influenced by p. We conclude that a gap still exists between the theoretical complexity and practical behavior of the algorithm.

Operations Research Proceedings, 2005

ABSTRACT We investigate the robust shortest path problem using the robust linear optimization met... more ABSTRACT We investigate the robust shortest path problem using the robust linear optimization methodology as proposed by Ben-Tal and Nemirovski. We discuss two types of uncertainty, namely, box uncertainty and ellipsoidal uncertainty. In case of box uncertainty, the robust counterpart is simple. It is a shortest path problem with the original arc lengths replaced by their upper bounds. When dealing with ellipsoidal uncertainty, we obtain a conic quadratic optimization problem with binary variables. We present an example to show that a subpath of a robust shortest path is not necessarily a robust shortest path.

SIAM Journal on Optimization, 2002

We introduce a new barrier function which is not a barrier function in the usual sense: it has fi... more We introduce a new barrier function which is not a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, its iteration bound, O √ n log n log n ε ,i s as good as it can be: it is the best known bound for large-update methods. The recently introduced

SIAM Journal on Optimization, 2004

Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) ... more Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on these kernel functions. In both the algorithm and its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of kernel function in the new class. We apply this scheme to seven specific kernel functions. Some of these functions are self-regular, and others are not. One of the functions differs from the others, and from all self-regular functions, in the sense that its growth term is linear. Iteration bounds for both large-and small-update methods are derived. It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely, O( √ n log n ε ). For large-update methods the best obtained bound is O( √ n(log n) log n ε ), which until now has been the best known bound for such methods.

RAIRO - Operations Research, 2008

ABSTRACT In this paper we present a generic primal-dual interior point methods (IPMs) for linear ... more ABSTRACT In this paper we present a generic primal-dual interior point methods (IPMs) for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. The proposed kernel function does not satisfy all the conditions proposed in [2]. We show that the corresponding large-update algorithm improves the iteration complexity with a factor n 1 6 when compared with the method based on the use of the classical logarithmic barrier function. For small-update interior point methods the iteration bound is O ( √ n log n), which is currently the best-known bound for primal-dual ɛ IPMs.

IEEE Transactions on Information Theory, 1981

A well-known theorem of MacWilliams states that b is the distribution of the dual code Cl wheneve... more A well-known theorem of MacWilliams states that b is the distribution of the dual code Cl whenever the code C is line?.

We propose a strategy for building up the linear program while using a logarithmic barrier method... more We propose a strategy for building up the linear program while using a logarithmic barrier method.The method starts with a (small) subset of the dual constraints, and follows the correspondingcentral path until the iterate is close to (or violates) one of the constraints, which is in turn addedto the current system. This process is repeated until an optimal solution is

The frequency assignment problem FAP is the problem of assigning frequencies to transmission link... more The frequency assignment problem FAP is the problem of assigning frequencies to transmission links such that no interference between signals occurs. This implies distance constraints between assigned frequencies of links. The objective is to minimize the number of used frequencies. We present a n i n teger linear programming formulation that is closely related to the vertex packing problem. Although the size of this formulation is an order of magnitude larger than the underlying network of links, we use the integer linear programming formulation within a branch-and-cut algorithm. This algorithm employs problem speci c and generic techniques such as reduction methods, primal heuristics, and branching rules to obtain optimal solutions. We report on computational experience with real-life instances.

SIAM Journal on Optimization, 1997

In this paper the new polynomial affine scaling algorithm of Jansen, Roos andTerlaky for LP is ex... more In this paper the new polynomial affine scaling algorithm of Jansen, Roos andTerlaky for LP is extended to PSD linear complementarity problems. The algorithmis immediately further generalized to allow higher order scaling. These algorithmsare also new for the LP case. The analysis is based on Ling's proof for the LP case,hence allows an arbitrary interior feasible pair to start with.

Optimization, 1994

The literature in the field of interior point methods for linear programming has been almost excl... more The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Güler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple approach which is based on an embedding of the primal problem and its dual

Operations Research Letters, 1997

The formulation of interior point algorithms for semide nite programming has become an active res... more The formulation of interior point algorithms for semide nite programming has become an active research area, following the success of the methods for large{ scale linear programming. Many interior point methods for linear programming have now been extended to the more general semide nite case, but the initialization problem remained unsolved.

Mathematical Programming, 1997

In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming ... more In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen, Roos and Terlaky enhances an asymptotical O(f~n L) complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semi-definite linear complementarity problems.