# Questions tagged [stochastic-programming]

For questions about optimization problems in which one or more parameters are stochastic, with known probability distributions or discrete scenarios. The goal is often, but not always, to minimize or maximize the expected value of the objective function.

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### 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 ...
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### 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 $\xi_1$...
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### How can I calculate the UB if I'm using Fischetti's Alternative Cut Generation for Benders'?

Assuming I have a stochastic MIP that I want to minimize using Benders' Decomposition. With the objective of the MP being: $$Minimize \; cx + \frac{1}{\lvert S \rvert} \sum_{s \in S} \theta_s$$ ...
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1 vote
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### 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 ...
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### How to deal with performance bottlenecks in Stochastic Vehicle Routing Problem with Benders' decomposition?

I've been working on solving a stochastic vehicle routing problem using Benders' decomposition with CPLEX in C++. Initially, my implementation struggled with larger instances, but I've made ...
• 317
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### 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. $$\inf_{\mathbb{P}\in\mathcal{D}^{WD}} \mathbb{P}\{\|\mathbf{A\zeta-b}\|_2 \leq 1\} \geq 1-\epsilon$$ ...
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 ...
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### How to pass the values of a variable of the first model to a parameter of the second model?

I am working on a scheduling problem which is NP-hard problem. Therefore, I decided to implement two-stage strategy to speed up the solution process. I need to pass the values of a variable from the ...
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### 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?
1 vote
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,...
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### What's the best way to speed up Benders Decomposition for a stochastic vehicle routing problem?

Currently I am working on an implementation of Benders Decomposition that solves a stochastic vehicle routing problem with synchronisation constraints. Sadly, at the moment it is not performing fast ...
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1 vote
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### Benchmark problems for Benders Decomposition

We are implementing a scheduling model using Benders Decomposition. Does someone know of any existing implementation of Benders or any repositories that contain continuous or integer problems solved ...
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1 vote
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### 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....
1 vote
452 views

### Understanding the L-shaped-method and the different variants of it

I am currently trying to understand the integer L-shaped-method/stochastic version of Benders Decomposition because I have practical problem MIP that is stochastic and thus has very good decomposition ...
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1 vote
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### How do I check convergence in stochastic Benders?

So in the deterministic version of Benders, the main process works like this: I initialize my x-vector (Integer variables from the master problem) and solve the dual subproblem (SP). I add an ...
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1 vote
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### Recommendations to understand stochastic version of Benders?

I successfully understood and implemented a Benders algorithm for the deterministic version of the problem I am studying. However, I have some issues now diving into the stochastic model. I have the ...
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### Tips for implementing Benders Decomposition in C++?

currently I am working on implementing Benders Decomposition for a large-scale stochastic MIP in C++ using the CPLEX Solver. I've been spending the last couple of months learning the programming ...
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### 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 ...
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### Deriving a lower bound for a two-stage stochastic problem

Assume an inventory stochastic optimization problem in the following form: $$\min\limits_{x\in X} c^\top x + \mathbb{E}_{\mathbb{\xi}}[\mathcal{Q}(x, \xi)]$$ Demand is the uncertain parameter, and is ...
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### Is Benders decomposition and the L-shaped method the same algorithm?

I've been studying the Benders decomposition method to solve stochastic integer problems. I've also stumbled across papers using a so called L-shaped-algorithm which also divides into master problem ...
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### 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 ...
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### 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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### How to find the maximizing number of expected delivered units of a probabilistic minimum cost flow problem?

I am a Bitcoin / Lightning Network open source developer and researcher and new here but very active on the sister site. In the context of my research I discovered the field of operations research ...
632 views

### Robust Optimization in Gurobi

I have a research problem where my Mixed Integer Linear Program has data that follow probability distributions. I am approaching this by creating some m instances through realizations of these random ...
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### Create constraint only for a recourse problem in stochastic linear programming

Question1: Northam Airlines is trying to decide how to partition a new plane for its Chicagoā Detroit route. The plane can seat $200$ economy class, passengers. A section can be partitioned off for ...
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### Control variables and cofounding effects in stochastic programming/,model predictive control/reinforcement learning

How can we be sure that confounding variables/control variables donāt pickup the effect our decisions w.r.t decision variables had on the actual control variable? Since the term control variable ...
144 views

### Decision dependent scenarios in Stochastic Programming

I am new to the field of stochastic programming and saw that the general formulation of a two-stage stochastic programming problem is given by: $$\min_{x\in X}\{ g(x)= f(x) + E_{\xi}[Q(x,\xi)]\}$$ ...
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509 views

### How do I solve a probability based knapsack problem?

I am looking to maximize my probability of reaching a given target value, by creating multiple groups of 6 items with different means and variances that stay within a weight limit. This is an example ...
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1 vote
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
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### Resource which explain different stochastic method with some intuition

Hope my question fit this community. I have taken Stochastic Optimization course (2 credits). The course content are: Deterministic VS Stochastic Linear Program Two-Stage Recourse Problem Multi-Stage ...
• 157
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### Can Expected value of perfect information be zero?

I tried to find the Expected Value of Perfect Information (EVPI) and the Value of the Stochastic Solution (VSS) of the following problem. Consider a farmer who specializes in raising wheat, corn, and ...
• 157
113 views

### Use of variance in job ordering

For a job-shop-like problem I have some constraints of this form: $$c_i \geq s_i + d_{iom} z_{iom}$$ $z_{iom}$ is a binary enabler, $d_{iom}$ is the delay on job $i$ for operation $o$ done by ...
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### 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 ...
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764 views

### Difference between "Online Optimization" and "Stochastic Optimization"/"Robust Optimization"?

I just came across the notion of Online optimization (I got a look on Wikipedia page and some other webpages), but it was not enough for me and I am looking for a more elaborated comparison, namely in ...
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### 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 ...
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### Can we simplify (perhaps linearize) this constraint?

We are dealing with a stochastic model and one of the constraints is \begin{align} y_j=\frac{\sum_{i \in I}\sum_{k \in K}\mathbb{E}\left[X_{ik}^2\right]x^k_{ij}}{\sum_{i \in I} \sum_{k \in K} \mathbb{...
413 views

### Optimality in L Shaped or Bender Decomposition

I was working on solving a two-stage stochastic problem using L Shaped method (Benders Decomposition). I have discussed the model here: Stochastic Facility Location Model. Do the single-cut/ multi-cut ...
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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{...
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### 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 ...
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291 views

### How to solve Stochastic Dynamic Program with huge state space?

I am modelling a stochastic dynamic program but because I need to store all information related to former sales, the state of the dynamic program increases and potentially it can growth so much which ...
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### Stochastic Facility Location Model

I am solving a stochastic facility location model using Benders decomposition (L-shaped algorithm). In each scenario, I want to allocate demands from origin to a fixed number of closest open ...
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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): $$\frac{1}{N}\sum_{i=1}^N f(x,\xi^i).$$...