Dual Bounds for Periodical Stochastic Programs

2022 ◽  
Author(s):  
Alexander Shapiro ◽  
Yi Cheng
Keyword(s):  
Upper Bound ◽  
Numerical Experiments ◽  
Dual Problem ◽  
Infinite Horizon ◽  
Stochastic Program ◽  
Discount Factor ◽  
Stochastic Programs ◽  
Multistage Stochastic Program ◽  
Optimal Value ◽  
Dual Bounds

A construction of the dual of a periodical formulation of infinite-horizon linear stochastic programs with a discount factor is discussed. The dual problem is used for computing a deterministic upper bound for the optimal value of the considered multistage stochastic program. Numerical experiments demonstrate behavior of that upper bound, especially when the discount factor is close to one.

The Stackelberg Games of Water Extraction with Myopic Agents

2021 ◽  
Author(s):  
Alain Jean-Marie ◽  
Mabel Tidball ◽  
Víctor Bucarey López
Keyword(s):  
Stackelberg Game ◽  
Infinite Horizon ◽  
Dynamic Game ◽  
Water Extraction ◽  
Discount Factor ◽  
Stackelberg Games ◽  
Groundwater Extraction ◽  
Strategic Interactions ◽  
Water Users ◽  
Threshold Policies

We consider a discrete-time, infinite-horizon dynamic game of groundwater extraction. A Water Agency charges an extraction cost to water users and controls the marginal extraction cost so that it depends not only on the level of groundwater but also on total water extraction (through a parameter [Formula: see text] that represents the degree of strategic interactions between water users) and on rainfall (through parameter [Formula: see text]). The water users are selfish and myopic, and the goal of the agency is to give them incentives so as to improve their total discounted welfare. We look at this problem in several situations. In the first situation, the parameters [Formula: see text] and [Formula: see text] are considered to be fixed over time. The first result shows that when the Water Agency is patient (the discount factor tends to 1), the optimal marginal extraction cost asks for strategic interactions between agents. The contrary holds for a discount factor near 0. In a second situation, we look at the dynamic Stackelberg game where the Agency decides at each time what cost parameter they must announce. We study theoretically and numerically the solution to this problem. Simulations illustrate the possibility that threshold policies are good candidates for optimal policies.

scholarly journals A Global Optimization Algorithm for Sum of Linear Ratios Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yuelin Gao ◽  
Siqiao Jin
Keyword(s):  
Global Optimization ◽  
Lower Bound ◽  
Programming Problem ◽  
Branch And Bound ◽  
Numerical Experiments ◽  
Original Problem ◽  
Bilinear Programming ◽  
Global Optimization Algorithm ◽  
Product Function ◽  
Optimal Value

We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

scholarly journals Measures of information in multistage stochastic programming

2012 ◽  
Author(s):  
Franceska Maggioni ◽  
Elisabetta Allevi ◽  
Marida Bertocchi
Keyword(s):  
Transportation Problem ◽  
Value Of Information ◽  
Optimal Solution ◽  
Stochastic Program ◽  
Multistage Stochastic Programming ◽  
Rolling Horizon ◽  
Stochastic Programs ◽  
Multistage Stochastic Programs ◽  
Over Time

Multistage stochastic programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. Providing bounds for their optimal solution may help in evaluating whether it is worth the additional computations for the stochastic program versus simplified approaches. In this paper we present a summary of the results in [22] where we generalize the value of information gained from deterministic, pair solution and rolling-horizon approximation in the two-stage case to the multistage stochastic formulation. Numerical results on a case study related to a simple transportation problem illustrate the described relationships.

AN IMPROVED REDUCTION METHOD FOR THE ROBUST OPTIMIZATION OF THE ASSIGNMENT PROBLEM

2012 ◽  
Vol 29 (05) ◽  
pp. 1250032 ◽  
Author(s):  
BYUNGJUN YOU ◽  
DAISUKE YOKOYA ◽  
TAKEO YAMADA
Keyword(s):  
Lower Bound ◽  
Exact Solutions ◽  
Upper Bound ◽  
Assignment Problem ◽  
Numerical Experiments ◽  
Reduction Method ◽  
Computation Time ◽  
Approximate Solutions ◽  
Cpu Time ◽  
Surrogate Relaxation

We are concerned with a variation of the assignment problem, where the assignment costs differ under different scenarios. We give a surrogate relaxation approach to derive a lower bound and an upper bound quickly, and show that the pegging test known for zero–one programming problems is also applicable to this problem. Next, we discuss how the computation time for pegging can be shortened by taking the special structure of the assignment problem into account. Finally, through numerical experiments we show that the developed method finds exact solutions for instances with small number of scenarios in relatively small CPU time, and good approximate solutions in case of many scenarios.

scholarly journals Efficient Simulation for the Maximum of Infinite Horizon Discrete-Time Gaussian Processes

2011 ◽  
Vol 48 (02) ◽  
pp. 467-489
Author(s):  
Jose Blanchet ◽  
Chenxin Li
Keyword(s):  
Relative Error ◽  
Numerical Experiments ◽  
Infinite Horizon ◽  
Sampling Procedure ◽  
Remarkable Feature ◽  
Efficient Simulation ◽  
Bounded Number ◽  
Great Generality ◽  
High Level ◽  
Positive Integers

We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, sayb. In great generality such a probability converges to 0 exponentially fast in a power ofb. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (inb) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hittingbin finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number,n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations asn↗ ∞. A remarkable feature of TBS is that it isnotbased on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.

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