Unanswered Questions
28 questions with no upvoted or accepted answers
9
votes
0
answers
230
views
Solving large-scale stochastic mixed integer program
What are some methods or algorithms for solving a large-scale stochastic mixed-integer optimization problem that runs on an hourly dataset for a year? Do we employ some kind of decomposition? (the ...
- 1,687
6
votes
0
answers
226
views
Airline revenue management re-solving problem
I am considering a bid prices (shadow price of the capacity constraint) problem (from Chen, L. and Homem-de Mello, T. (2009)., page 14) where the acceptable classes for booking requests for ...
- 165
6
votes
0
answers
95
views
Sample Average Approximation vs. Numerical Integration
In the sense of the calculation of the expected value of objective functions, we have two choices to evaluate the value; 1. Sample Average Approximation (SAA):
$$
\frac{1}{N}\sum_{i=1}^N f(x,\xi^i).
$$...
- 155
5
votes
0
answers
154
views
Chance constrained optimization - interpretation
Suppose that we have a stochastic vector $\psi$ and $S$ realisations of $\psi$ given by $\psi_1,\dots,\psi_S$ with equal probability of occurrence. In addition, we have constraints of the form
\begin{...
- 2,463
4
votes
0
answers
164
views
Stochastic optimization for inventory management
The deterministic problem is to minimize operational cost subject to constraints in demand, supply and capacity. The ordering policy is periodic review, order-up-to.
The stochastic version of the ...
- 1,895
4
votes
0
answers
54
views
How to find range of values for the first-stage decisions resulting in the same cuts in two-stage stochastic programming?
Suppose we have a two-stage stochastic program as follows:
\begin{equation}
\begin{split}
\min \ & c^Tx + \mathbb{E}_\xi[Q(x,\xi)] \\
& \text{where}\\
&Q(x,\xi)=\min q(\xi)^Ty\\
&Tx+...
- 2,160
3
votes
0
answers
50
views
Control variables and cofounding effects in stochastic programming/,model predictive control/reinforcement learning
How can we be sure that confounding variables/control variables don’t pickup the effect our decisions w.r.t decision variables had on the actual control variable?
Since the term control variable ...
3
votes
0
answers
89
views
Derivative of sup(max) functions in distributionally robust optimization
In the distributionally robust optimization problem
\begin{aligned}
\min_{x\in X}\sup_{P\in\mathfrak{P}}\mathbb{E}_P[f(x,\xi)],
\end{aligned}
where $f:\mathbb{R}^n\to\mathbb{R}$ and $P$ is a ...
- 155
2
votes
0
answers
86
views
Reformulate the deterministic equivalent model as an Expected Value problem
Given an optimization problem as follows:
$$
\begin{array}{cc}
\operatorname{Max} Z=3 x_{1}+9 x_{2}-2 y_{1}-4 y_{2} \\
\text { subject to, } y_{1}+y_{2}=15 \\
5 x_{1}+2 x_{2} \leq 10 \\
x_{1}, x_{2}, ...
- 157
2
votes
0
answers
100
views
Two-stage stochastic with non-linear recourse
I am working on a two-stage facility location problem as I described in this question.
I am solving it with the L-shaped method (Benders decomposition). The cost value between each $(i,j)$ is a ...
- 5,442
2
votes
0
answers
312
views
What is the intuition behind Progressive hedging algorithm?
I am reading some papers about PHA to solve multi-stage stochastic programming, but I think it is not still clear to me. This is my understanding and I would be thankful to know if it is correct or ...
- 2,160
2
votes
0
answers
812
views
How to write nonanticipativity constraints?
In Multi-stage stochastic programming, we write the constraints that for scenarios $s$ and $s^{\prime}$ which have the same trajectory up to time $t$, should take the same value. That is,
$$
x_{t,s} =...
- 2,160
2
votes
0
answers
63
views
Decision-making algorithm for dynamic load balancing
I'm researching a subject of balancing the load between two black-box systems (with some twists). I thought that I could record latest response time log from each of those systems and process such a ...
1
vote
0
answers
37
views
Using the Alternative Cut Generation Problem in Benders, why do I get different results?
I am using Benders' Decomposition to solve a stochastic MIP.
To improve cut selection, I implemented the Alternative Cut Generation Problem as proposed by Fischetti et al. (2010).
I will summarize the ...
- 327
1
vote
0
answers
21
views
Building a CapEx portfolio using mathematical optimization
Let's say you have a set of potential capital projects $C$, each defined by an up-front investment $c_i$ and random payoff (say, NPV) $P_i(\omega)$, where $\omega \in \Omega$ is a point in a sample ...
- 111
1
vote
0
answers
41
views
How would the Contextual Stochastic Optimization framework be applied to a bilevel problem whose uncertain parameters lie in the inner problem?
Although I only heard about Contextual Stochastic Optimization (CSO) a few months ago, I know now the excitement has been going on for a while. I'm not sure if the idea of CSO has been around for long,...
- 1,895
1
vote
0
answers
58
views
Distributionally Robust Stochastic Programming - Help with derivation
I've been working through this book on robust optimization of electric energy systems, and in particular chapter 4 on distributionally robust optimization. In following the derivation of section 4.2.1....
- 5,874
1
vote
0
answers
38
views
Multicollinearity w.r.t decisions in optimal control/reinforcement learning learning/resource allocation problem
Consider the following optimization/control problem:
We aim to maximize the cumulative reward $R$ during the horizon $H$ by every day allocating a portion of total budget $B$ to our two different ...
1
vote
0
answers
71
views
Multi-Stage Stochastic Decomposition
I have a multi-stage model with both binary and continuous first-stage investment variables and continuous operational next-stage variables:
$$
\sum_{s} \rho_{s} \left[ x_{s} + y_{s} + \sum_{t}(y^{op}...
- 113
1
vote
0
answers
68
views
Scenario Tree Construction in Multi-Stage Stochastic Programming
I gonna use the approach used in this document. Suppose there are $T$ stages and uncertain parameter is $\xi_{t}, \quad t \in \{1,2,\dots,T\}$.
In this algorithm, it is required to calculate the ...
- 2,160
1
vote
0
answers
117
views
How do I find the extreme rays and points for a stochastic programming problem
I have the following 2 stage Stochastic Programming program:
\begin{align}\min_x& \quad x+\sum_{s=1}^{3}p_sQ_s(x)\\\text{s.t.}&\quad x\in\Bbb R\\&\quad Q_s(x)=\min\left[\begin{pmatrix}1&...
- 149
0
votes
0
answers
26
views
Understanding different norms in the p-Wasserstein distance
The generalized p-Wasserstein distance, for $p\geq 1$, is given by
$$d_W(Q_1,Q_2):=inf \left\{\int_{\Xi_2}||\xi_1-\xi_2||^p \Pi(d\xi_1,d\xi_2)\right\}$$
where $\Pi$ is the joint distribution of $\xi_1$...
- 31
0
votes
0
answers
20
views
Supremum of a probabilistic function with ambiguity distribution set using Wasserstein metric
There is a proof of how to derive distributionally robust chance constraints with ellipsoid bound.
$$\inf_{\mathbb{P}\in\mathcal{D}^{WD}} \mathbb{P}\{\|\mathbf{A\zeta-b}\|_2 \leq 1\} \geq 1-\epsilon$$
...
0
votes
0
answers
56
views
How to initialize a parameter (belonging to the first stage model) in a two stage model, taking its value from second stage model?
I am working on a two stage approach in order to reduce the complexity of a scheduling model which is an NP-hard problem. I have to implement a while loop in order to repeat solving the models in case ...
- 131
0
votes
0
answers
46
views
monte carlo for a selection problem
What type of Monte Carlo simulation is suitable for solving this problem? Also, how can we select results after simulation to guarantee feasibility?
- 8,677
0
votes
0
answers
69
views
Simplex algorithm for stochastic constraints?
The OR-Notes by J E Beasley states:
Hence the problem:
minimise 5x+6y
subject to:
Prob(a1x + a2y >= 3) >= 1-alpha
x,y >= 0
...
- 1,852
0
votes
0
answers
84
views
Resource allocation problem - RL or stochastic optimization?
I am currently working on a resource allocation problem and I am uncertain about which field of stochastic optimization and reinforcement learning encompasses this particular problem.
The objective is ...
- 1,895
0
votes
1
answer
99
views
Theorem proving Stochastic Optimization for Unit Commitment always better than deterministic solution
I'm trying to recall a theorem that I was taught in grad school, but can't remember the name of the theorem. We were learning about different methods to solve unit committment and economic dispatch ...
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