9

The following is purely personal opinion. I would say a (substantial) majority of non-academic optimization problems do not involve any of the methods you listed, for a number of reasons. "Better is the enemy of good enough." Using fixed, plausible values for parameters and ignoring uncertainty often produce answers that are good enough for ...


4

To solve stochastic programming models with integer recourse, there are some methods. Most stochastic programming textbooks cover these methods. For example, chapter 7 of Introduction to Stochastic Programming by Birge and Louveux covers these techniques. In particular, I suggest either using the integer L-shaped method or the progressive hedging algorithm (...


3

How to generate a multivariate Gaussian? It must be answered somewhere on Cross Validated, but I cannot find it now, some comments at https://stats.stackexchange.com/questions/341805/are-mvrnorm-in-mass-r-package-and-rmvn-in-mgcv-r-package-equivalent/341808#341808. Let $X \sim \mathcal{N}(\mu, \Sigma)$ and $\epsilon \sim \mathcal{N}(0,I)$. Then we can ...


3

Yes. Assuming that $\alpha$, $\beta$ and $\gamma$ in the text are the same as $a$, $b$ and $g$ in the model, then $b_i$ should be $b_i y_i$.


3

Whether maximizing expected winnings is an appropriate solution depends on the underlying problem and, crucially, the consumer of the solution (the problem "owner"). Two major considerations jump out at me. One is whether the owner requires a certain minimum payout with a certain probability, in which case you may need to use chance constrained ...


2

CPLEX treats certain small values as negligible for purposes of constraint satisfaction (including satisfying integrality constraints). That does not mean it automatically rounds small values to zero. If the coefficient 1.4210854715202e-14 in your cut is the value of a dual variable from a subproblem, it is up to you to decide whether to round it to zero ...


2

I'm still not sure I understand the question, but I'll suggest an answer to what I think is being asked. I'm going to assume that an a priori upper bound $O$ exists for $o_t$. To keep what follows somewhat compact, I'm going to assume that $\mathcal{M} = \lbrace 20, 21, 22, 23\rbrace$ and $O=6$. To start, for each $t\in T$ generate a random sample of $O$ ...


2

Standard assumption: we write a general second-stage problem as $Q(x,\xi)=\min_y\{q^\top y\mid Wy=h-Tx\}$ and $\xi=(q,W,T,h)$. In your example $W$ and $T$ are fixed (not stochastic) while $q=(\xi_1,1)$ and $h=(1,\xi_2)$ are stochastic. Now, $Q(x,\xi)$ is convex in $h$, $T$ and $x$ but is concave in $q$. So, saying that $Q(x,\xi)$ is, in general, convex in $x$...


2

You can try a master problem of the form \begin{alignat*}{1} \min & \quad \gamma\\ \textrm{s.t.} & \quad \sum_{s=1}^{S}P_{s}Y_{s}\le\alpha\\ & \quad \gamma\ge\gamma_{T}\left[\sum_{s\in T}(1-Y_{s})-|T|+1\right]\quad\forall T\in\mathcal{T}\\ & \quad Y_{s}\in\left\{ 0,1\right\} \quad\forall s\in\left\{ 1,\dots,S\right\} \end{alignat*} where $\...


2

The issue you are describing has to do with the necessity of accounting for both short- and long-term dynamics in a decision problem under uncertainty, or in general uncertainty at different levels of resolution. There are two issues here. The practical implementation of a stochastic program lives on a scenario tree. So the first issue is how to arrange ...


2

Let's make it more clear. Stochastic Programming is not Stochastic Optimization. When you say Stochastic Programming the above answer of @Larry is explaining more about "stochastic programming". This is when your problem of study has some uncertainty (e.g. demand uncertainty, arrival times uncertainty). The robust optimization sets with Stochastic ...


1

Suppose there are three scenarios $i$ in the second stage and two time steps $t$. With $$ x_1 = \begin{pmatrix} x_{11} \\ x_{12} \\ \end{pmatrix},\quad x_2 = \begin{pmatrix} x_{21} \\ x_{22} \\ \end{pmatrix},\quad x_3 = \begin{pmatrix} x_{31} \\ x_{32} \\ \end{pmatrix}, $$ one obvious choice would be $$ H_1 = \begin{...


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