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A linear-time algorithm for concave one-dimensional dynamic programming

Zvi Galil, Kunsoo Park
1990 Information Processing Letters  
The one-dimensional dynamic programming problem is defined as follows: given a real-valued function w(i,j) for integers 0 constant time.  ...  A[i, j] denotes the element in the ith row and the jth column.  ...  In the concave one-dimensional dynamic programming problem w is concave as defined above. A condition closely related to the quadrangle inequality was introduced by Aggarwal et al.  ... 
doi:10.1016/0020-0190(90)90215-j fatcat:qkzbv6tn2jhwjf5a7u5u5xhaca

Page 4806 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
process in order to pro- duce a tight higher dimensional linear programming relaxation.  ...  Even the best concave minimization algorithms can, in general, not solve problems with more than 50 variables in a reasonable time.  ... 

Page 7466 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
algorithm for the solution of stochastic dynamic minimax problems.  ...  The latter can be converted into m one-dimensional linear integer problems, which are to be solved. Section 2 includes the description of the pro- posed algorithm.  ... 

An Optimal Approximate Dynamic Programming Algorithm for Concave, Scalar Storage Problems With Vector-Valued Controls

Juliana Nascimento, Warren B. Powell
2013 IEEE Transactions on Automatic Control  
We prove convergence of an approximate dynamic programming algorithm for a class of high-dimensional stochastic control problems linked by a scalar storage device, given a technical condition.  ...  Our algorithm exploits the natural concavity of the problem to avoid any need for explicit exploration policies. Index Terms-Approximate dynamic programming, resource allocation, storage.  ...  INTRODUCTION W E propose an approximate dynamic programming algorithm that is characterized by multi-dimensional (and potentially high-dimensional) controls, but where each time period is linked by a single  ... 
doi:10.1109/tac.2013.2272973 fatcat:rf3fjrf745dprjhwkyuxwuy6bi

An Optimal Approximate Dynamic Programming Algorithm for the Lagged Asset Acquisition Problem

Juliana M. Nascimento, Warren B. Powell
2009 Mathematics of Operations Research  
We prove convergence of an approximate dynamic programming algorithm for a class of high-dimensional stochastic control problems linked by a scalar storage device, given a technical condition.  ...  Our algorithm exploits the natural concavity of the problem to avoid any need for explicit exploration policies. Index Terms-Approximate dynamic programming, resource allocation, storage.  ...  INTRODUCTION W E propose an approximate dynamic programming algorithm that is characterized by multi-dimensional (and potentially high-dimensional) controls, but where each time period is linked by a single  ... 
doi:10.1287/moor.1080.0360 fatcat:lfyfzqmlunc45mbphraa5mczlu

Page 2813 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews  
For the concave case, we present a linear time algorithm for on-line searching in totally monotone matrices which is a generalization of the on-line one-dimensional prob- lem.  ...  Our algorithm uses as a module an algorithm for solving a certain on-line one-dimensional dynamic programming problem.  ... 

Page 1179 of Mathematical Reviews Vol. , Issue 95b [page]

1995 Mathematical Reviews  
Furthermore, the duality results presented here are dynamic gener- alizations of finite-dimensional nonlinear programming problems recently explored.”  ...  Summary: “In this paper, the Iri-Imai algorithm for solving linear and cdOnvex quadratic programming is extended to solve some other smooth convex programming problems.  ... 

Page 1062 of Mathematical Reviews Vol. 52, Issue 3 [page]

1976 Mathematical Reviews  
52 #/7549-7558 Author’s summary: “In this paper we describe two algorithms in which a maximisation program of a numerical concave or quasi- concave function in a convex set CC R" is replaced by a sequence  ...  The authors present an algorithm for solving a nonlinear program- ming problem whose objective function is the product of linear functions and whose constraints are linear.  ... 

Approximation Algorithms for Optimization of Combinatorial Dynamical Systems [article]

Insoon Yang, Samuel A. Burden, Ram Rajagopal, S. Shankar Sastry, Claire J. Tomlin
2015 arXiv   pre-print
The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system.  ...  This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables.  ...  Therefore, existing polynomial-time exact and approximation algorithms for 0-1 linear programs can be employed, as shown in Section IV.  ... 
arXiv:1409.7861v2 fatcat:zpureo77kjcjhdzjka6axiidle

Linear Regression without Correspondences via Concave Minimization [article]

Liangzu Peng, Manolis C. Tsakiris
2020 arXiv   pre-print
The resulting algorithm outperforms state-of-the-art methods for fully shuffled data and remains tractable for up to 8-dimensional signals, an untouched regime in prior work.  ...  Linear regression without correspondences concerns the recovery of a signal in the linear regression setting, where the correspondences between the observations and the linear functionals are unknown.  ...  Finally, a working algorithm for unlabeled sensing was built in [4] to globally optimize (2) by combining branch-and-bound and dynamic programming to repeatedly solve a one-dimensional linear assignment  ... 
arXiv:2003.07706v1 fatcat:xtt3hvyaxjdizakp3mw2hiby7u

An Optimal ADP Algorithm for a High-Dimensional Stochastic Control Problem

Juliana Nascimento, Warren Powell
2007 2007 IEEE International Symposium on Approximate Dynamic Programming and Reinforcement Learning  
We propose a provably optimal approximate dynamic programming algorithm for a class of multistage stochastic problems, taking into account that the probability distribution of the underlying stochastic  ...  The algorithm and its proof of convergence rely on the fact that the optimal value functions of the problems within the problem class are concave and piecewise linear.  ...  CONCLUSIONS We proposed an approximate dynamic programming algorithm for a class of multistage stochastic control problems.  ... 
doi:10.1109/adprl.2007.368169 fatcat:kl7lq6iitrbjlaiyfczknpdode

Approximate Dynamic Programming for High-Dimensional Resource Allocation Problems [chapter]

2009 Handbook of Learning and Approximate Dynamic Programming  
These can be formulated as dynamic programs, but typically exhibit high dimensional state, action and outcome variables (the three curses of dimensionality).  ...  For example, we have worked on problems where the dimensionality of these variables is in the ten thousand to one million range.  ...  Section III describes an algorithmic strategy using the principles of approximate dynamic programming.  ... 
doi:10.1109/9780470544785.ch10 fatcat:gevi3n4r5vcsjfkbih6jwziqwq

Algorithms for minimum-cost paths in time-dependent networks with waiting policies

Brian C. Dean
2004 Networks  
In discrete time, time-dependent shortest path problems with waiting constraints can be optimally solved by straightforward dynamic programming algorithms; however, for some waiting policies these algorithms  ...  In this paper we survey several broad classes of waiting policies and show how techniques for speeding up dynamic programming can be effectively applied to obtain practical algorithms for these different  ...  We utilize techniques for speeding up certain classes of "one-dimensional" dynamic programming algorithms that were initially motivated by a host of problems from various fields, most notably by sequence  ... 
doi:10.1002/net.20013 fatcat:shxlbnj5mbgcxp5h7wgjvdvny4

Rapid dynamic programming algorithms for RNA secondary structure

Michael S Waterman, Temple F Smith
1986 Advances in Applied Mathematics  
In this paper new dynamic programming algorithms are presented which reduce the required computation. The first polynomial time algorithm is given for predicting general secondary structure. Q  ...  Dynamic programming methods are currently the most useful computer technique but are frequently very expensive in running time.  ...  ACKNOWLEDGMENT The authors are grateful to Michael Zuker for his useful comments about this work.  ... 
doi:10.1016/0196-8858(86)90025-4 fatcat:vzvzdaombjgj5ahpw2vmms5t6q

Page 3218 of Mathematical Reviews Vol. , Issue 85g [page]

1985 Mathematical Reviews  
85g:90081 Cooper, Mary W. (1-SMU); Farhangian, Keyvan (1-SMU) A dynamic programming algorithm for multiple-choice constraints. Comput. Math. Appl. 10 (1984), no. 3, 279-282.  ...  Consequently, if a1,a2,---,a, are fixed, there are no more than (n—1)[]"_, a; problems for which a polynomial time algorithm is unknown.”  ... 
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