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

1990
*
Information Processing Letters
*

The

doi:10.1016/0020-0190(90)90215-j
fatcat:qkzbv6tn2jhwjf5a7u5u5xhaca
*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. ...##
###
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

2013
*
IEEE Transactions on Automatic Control
*

We prove convergence of an approximate

doi:10.1109/tac.2013.2272973
fatcat:rf3fjrf745dprjhwkyuxwuy6bi
*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 ...##
###
An Optimal Approximate Dynamic Programming Algorithm for the Lagged Asset Acquisition Problem

2009
*
Mathematics of Operations Research
*

We prove convergence of an approximate

doi:10.1287/moor.1080.0360
fatcat:lfyfzqmlunc45mbphraa5mczlu
*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 ...##
###
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]

2015
*
arXiv
*
pre-print

The approximate problem is

arXiv:1409.7861v2
fatcat:zpureo77kjcjhdzjka6axiidle
*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. ...##
###
Linear Regression without Correspondences via Concave Minimization
[article]

2020
*
arXiv
*
pre-print

The resulting

arXiv:2003.07706v1
fatcat:xtt3hvyaxjdizakp3mw2hiby7u
*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 ...##
###
An Optimal ADP Algorithm for a High-Dimensional Stochastic Control Problem

2007
*
2007 IEEE International Symposium on Approximate Dynamic Programming and Reinforcement Learning
*

We propose

doi:10.1109/adprl.2007.368169
fatcat:kl7lq6iitrbjlaiyfczknpdode
*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. ...##
###
Approximate Dynamic Programming for High-Dimensional Resource Allocation Problems
[chapter]

2009
*
Handbook of Learning and Approximate Dynamic Programming
*

These can be formulated as

doi:10.1109/9780470544785.ch10
fatcat:gevi3n4r5vcsjfkbih6jwziqwq
*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*. ...##
###
Algorithms for minimum-cost paths in time-dependent networks with waiting policies

2004
*
Networks
*

In discrete

doi:10.1002/net.20013
fatcat:shxlbnj5mbgcxp5h7wgjvdvny4
*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 ...##
###
Rapid dynamic programming algorithms for RNA secondary structure

1986
*
Advances in Applied Mathematics
*

In this paper new

doi:10.1016/0196-8858(86)90025-4
fatcat:vzvzdaombjgj5ahpw2vmms5t6q
*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. ...##
###
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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