GenerateInfeasibilityReport(Log, xprob, constraintViolations, boundViolations)ĪpplyRepairInfeasResultToProblem(xprob, constraintViolations, boundViolations) GetInfeasibilityBreakers(xprob, x, slacks, constraintViolations, boundViolations, 0.000001) ' NOTE: For MIPs, use xprob.GetMIPSol instead ' Get the values of the breaker variables X prob.GetLpSol(x, slacks, Nothing, Nothing) Log.WriteLine( "Solution of the relaxed problem regarding the original objective is nonoptimal") Log.WriteLine( "Relaxed problem is unbounded") Log.WriteLine( "Relaxed problem is infeasible") X prob.RepairWeightedInfeas(Status, lrp_array, grp_array, lbp_array, ubp_array, "n", 0.001, "") Next ' Call Repairinfeas Dim Status As Integer = -1 Next For j = 0 To x prob.OriginalCols - 1 ' Allocate preference arrays Dim lrp_array(x prob.Rows) As Double Dim grp_array(x prob.Rows) As Double Dim lbp_array(x prob.Cols) As Double Dim ubp_array(x prob.Cols) As Double ' Allocate arrays for the solution and the infeasibility breakers Dim x(x prob.Cols) As Double Dim slacks(x prob.Rows) As Double Dim boundViolations(x prob.Cols) As Double Dim constraintViolations(x prob.Rows) As Double ' Set relaxation values Dim i As Integer, j As Integer For i = 0 To x prob.OriginalRows - 1 X prob.ReadProb(frmMain.sDataDirPath & "/" & sProblem) Xprob = New XPRSprob ' Forward messages to log But I suspect that it might be straightforward to have a direct construction, more direct than that from unwinding the proof of residuality, by exploiting the fact that whenever you have any ergodic system of zero entropy, such as irrational rotation on the unit circle, you obtain a zero entropy process just by observing the ergodic system through any finite partition.Const sProblem As String = "infeas.lp" Public Sub GetBreakers( ByVal Log As TextWriter)ĭim xprob As XPRSprob = Nothing Try ' Initialize optimizer XPRS.Init() That's the only proof that I know of for the existence of fully supported, ergodic, zero entropy measure on $\Sigma$. Such measures actually form a residual subset of the space of all invariant probability measures on $\Sigma$, for which see "ergodic theory on compact spaces" by Denker & Grillenberger & Sigmund. So you only need to construct an example of a fully supported, ergodic measure of zero entropy. Yet another way of producing a counter-example is to look inside the class of zero entropy processes because Markov measures always have positive entropy and if IIRC, this is also true for Gibbs measures.
Irreducible subshift code#
In fact, any time you have two different ergodic measures $\mu, \mu'$ on an mixing SFT projecting to the same measure $\pi\mu = \pi\mu'$ via some finite-to-one factor code onto a mixing SFT, the image $\pi\mu$ cannot be a Gibbs state in the sense of "R W"'s answer, let alone a Markov measure. It is probably one of the simplest counter-examples for your question.
Irreducible subshift full#
Following this terminology, we can say that the image measure $\pi\mu_p$, for $0 < p < 1/2$, in Example 2.9 in the preprint is a sofic, non-Markov measure on the full two-shift. In the preprint, as in many related literature, "Markov measure" is defined to be any invariant probability measure on any irreducible SFT (subshift of finite type) with full support and with n-step Markov property for some n, and "sofic measure" is defined to be images of such measures under factor codes. I'll refer to the arxiv v2 of the preprint. Another strict generalization I'd like to mention is (stationary) hidden Markov chains, for which see "Hidden Markov processes in the context of symbolic dynamics" by Boyle & Petersen. It feels like that question should have an easy answer, but somehow I don't get it.Īs "R W"s answer points out, Gibbs measures (of regular enough potentials, such as Holder continuous potentials) are a strict generalization of Markov measures. However, in the described set-up I could not find one. It is quite easy to construct counter-examples if I drop certain assumptions. Let $\Omega = \$ are the elements of a compatible stochastic matrix and $p$ is its unique unity stochastic eigenvector? It seems rather intuitive, however, I was not able to proof it yet: I have this question I have been struggling with for a while.
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