improved termination criteria text

bibliography/bibliography.bib

@@ -51,9 +51,11 @@ @string { a_chen_bo
 @string { a_cheng_edwin_tai_chiu = "Edwin Tai Chiu Cheng" }
 @string { a_chiong_raymond = "Raymond Chiong" }
 @string { a_chretienne_philippe = "Philippe Chr{\'e}tienne" }
+@string { a_clausen_anders = "Anders Clausen" }
 @string { a_coffman_edward_g = "Edward G.\ Coffman" }
 @string { a_cook_stephen_arthur = "Stephen Arthur Cook" }
 @string { a_cook_william_john = "William John Cook" }
+@string { a_crepinsek_matej = "Matej {\v{C}}repin{\v{s}}ek" }
 @string { a_davis_jim = "Jim Davis" }
 @string { a_de_causmaecker_patrick = "Patrick {De Causmaecker}" }
 @string { a_delorme_maxence = "Maxence Delorme" }
@@ -80,6 +82,7 @@ @string { a_garey_michael_r
 @string { a_geihs_kurt = "Kurt Geihs" }
 @string { a_gen_mitsuo = "Mitsuo Gen" }
 @string { a_gharbi_anis = "Anis Gharbi" }
+@string { a_ghoreishi_seyyedeh_newsha = "Seyyedeh Newsha Ghoreishi" }
 @string { a_giacobini_mario = "Mario Giacobini" }
 @string { a_gonzalez_arce_teofilo_francisco = "Teofilo Francisco {Gonzalez Arce}" }
 @string { a_gorldt_christian = "Christian Gorldt" }
@@ -106,6 +109,7 @@ @string { a_john_robert_i
 @string { a_johnson_david_stiffler = "David Stifler Johnson" }
 @string { a_johnson_e_l = "E.\ L.\ Johnson" }
 @string { a_johnson_selmer_martin = "Selmer Martin Johnson" }
+@string { a_jorgensen_bo_norregaard = "Bo N{\o}rregaard J{\o}rgensen" }
 @string { a_juenger_michael = "Michael J{\"u}nger" }
 @string { a_karp_richard_m = "Richard M.\ Karp" }
 @string { a_kendall_graham = "Graham Kendall" }
@@ -126,10 +130,12 @@ @string { a_lawler_eugene_leighhton
 @string { a_lawrence_stephen_r = "Stephen R.\ Lawrence" }
 @string { a_lenstra_jan_karel = "Jan Karel Lenstra" }
 @string { a_li_jiawei = "Jiawei Li" }
+@string { a_liu_shih_hsi = "Shih{-}Hsi Liu" }
 @string { a_liu_weichen = "Weichen Liu" }
 @string { a_liu_zhen = "Zhen Liu" }
 @string { a_lodi_anrea = "Andrea Lodi" }
 @string { a_machado_penousal = "Penousal Machado" }
+@string { a_madani_kurosh = "Kurosh Madani" }
 @string { a_maeda_tatsuo = "Tatsuo Maeda" }
 @string { a_manderick_bernard = "Bernard Manderick" }
 @string { a_manner_reinhard = "Reinhard M{\"a}nner" }
@@ -142,6 +148,7 @@ @string { a_mccormack_jon
 @string { a_meeran_sheik = "Sheik Meeran" }
 @string { a_mehta_sanjay_v = "Sanjay V.\ Mehta" }
 @string { a_merelo_guervos_juan_julian = "Juan Juli{\'a}n {Merelo Guerv{\'o}s}" }
+@string { a_mernik_marjan = "Marjan Mernik" }
 @string { a_michalewicz_matthew = "Matthew Michalewicz" }
 @string { a_michalewicz_zbigniew = "Zbigniew Michalewicz" }
 @string { a_michel_laurent = "Laurent Michel" }
@@ -155,6 +162,7 @@ @string { a_neri_ferrante
 @string { a_neumann_frank = "Frank Neumann" }
 @string { a_ng_chi_to_daniel = "Chi{-}To Daniel Ng" }
 @string { a_oneill_michael = "Michael {O'Neill}" }
+@string { a_oreilly_una_may = "Una{-}May O'Reilly" }
 @string { a_otto_boris = "Boris Otto" }
 @string { a_pardalos_panos_m = "Panos Miltiades Pardalos" }
 @string { a_pentek_tobias = "Tobias Pentek" }
@@ -167,12 +175,14 @@ @string { a_potts_chris_n
 @string { a_preuss_mike = "Mike Preu{\ss}" }
 @string { a_punnen_abraham_p = "Abraham P.\ Punnen" }
 @string { a_qu_rong = "Rong Qu" }
+@string { a_ravber_miha = "Miha Ravber" }
 @string { a_rechenberg_ingo = "Ingo Rechenberg" }
 @string { a_reinelt_gerhard = "Gerhard Reinelt" }
 @string { a_reinhard_kai = "Kai Reinhard" }
 @string { a_rinnooy_kan_a_h_k = "Alexander Hendrik George {Rinnooy Kan}" }
 @string { a_rothlauf_franz = "Franz Rothlauf" }
 @string { a_sabar_nasser = "Nasser Sabar" }
+@string { a_sabourin_christophe = "Christophe Sabourin" }
 @string { a_sahni_sartaj = "Sartaj Sahni" }
 @string { a_sannomiya_nobu = "Nobuo Sannomiya" }
 @string { a_sarli_michael = "Michael Sarli" }
@@ -205,6 +215,7 @@ @string { a_van_hoorn_jelke_jeron
 @string { a_vanden_berghe_greet = "Greet {Vanden Berghe}" }
 @string { a_vilim_petr = "Petr Vil{\'i}m" }
 @string { a_voigt_hans_michael = "Hans{-}Michael Voigt" }
+@string { a_warwick_kevin = "Kevin Warwick" }
 @string { a_watson_richard_a = "Richard A.\ Watson" }
 @string { a_weise_thomas = "Thomas Weise" }
 @string { a_werner_frank = "Frank Werner" }
@@ -244,6 +255,7 @@ @string { p_railway_technical_research_institute
 @string { p_sage_publications_inc = "{{SAGE} Publications Inc.}" }
 @string { p_sage_publications_ltd = "{{SAGE} Publications Ltd.}" }
 @string { p_scientific_research_publishing_inc = "{Scientific Research Publishing Inc. ({SCIRP})}"}
+@string { p_scitepress = "SciTePress, Science and Technology Publications, Lda" }
 @string { p_siam = "{Society for Industrial and Applied Mathematics Publications~({SIAM})}" }
 @string { p_springer_boston = "{Springer}" }
 @string { p_springer_international_publishing = "{Springer International Publishing~{AG}}" }
@@ -279,6 +291,8 @@ @string { l_israel_haifa
 @string { l_japan_kobe = "{{Kobe}, {Japan}}" }
 @string { l_japan_tokyo = "{{Tokyo}, {Japan}}"}
 @string { l_netherlands_amsterdam = "{{Amsterdam}, {The Netherlands}}" }
+@string { l_portugal_funchal = "{{Funchal}, {Madeira}, {Portugal}}" }
+@string { l_portugal_setubal = "{{Set{\'u}bal}, {Portugal}}"}
 @string { l_spain_barcelona = "{{Barcelona}, {Spain}}" }
 @string { l_spain_granada = "{{Granada}, {Spain}}" }
 @string { l_switzerland_cham = "{{Cham}, {Switzerland}}" }
@@ -332,6 +346,7 @@ @string { pa_sage_publications_inc
 @string { pa_sage_publications_ltd = l_uk_london }
 @string { pa_siam = l_usa_philadelphia }
 @string { pa_scientific_research_publishing_inc = l_usa_los_angeles }
+@string { pa_scitepress = l_portugal_setubal }
 @string { pa_springer_boston = l_usa_boston }
 @string { pa_springer_international_publishing = l_switzerland_cham }
 @string { pa_springer_nature_america = l_usa_new_york }
@@ -358,8 +373,21 @@ @xdata{c_gecco
  address = pa_acm,
 }
 
+@xdata{c_ijcci,
+ issn = {2184-3236},
+ publisher = p_scitepress,
+ address = pa_scitepress,
+}
+
 
 %% journals
+@xdata{j_asoc,
+ journal = {Applied Soft Computing {(ASOC)}},
+ issn = {1568-4946},
+ publisher = p_elsevier,
+ address = pa_elsevier
+}
+
 @xdata{j_cace,
  journal = {Computers {\&} Chemical Engineering},
  issn = {0098-1354},
@@ -923,6 +951,15 @@ @incollection{FT1963PLCOLJSSR
  pages = {225--251}
 }
 
+@inproceedings{GCJ2017TCIEAAS,
+ author = a_ghoreishi_seyyedeh_newsha # { and } # a_clausen_anders # { and } # a_jorgensen_bo_norregaard,
+ title = {Termination Criteria in Evolutionary Algorithms: {A} Survey},
+ pages = {373--384},
+ doi = {10.5220/0006577903730384},
+ crossref = {PROC2017IJCCI},
+ url = {https://pdfs.semanticscholar.org/05ca/33f534824b924d415623dce9eec420b33baf.pdf}
+}
+
 @book{GJ1979CAIAGTTTONC,
  author = a_garey_michael_r # { and } # a_johnson_david_stiffler,
  title = {Computers and Intractability: {A} Guide to the Theory of {NP}-Completeness},
@@ -1390,6 +1427,14 @@ @proceedings{PROC2016HICSS
  editor = a_bui_tung_x # { and } # a_sprague_jr_ralph_h,
 }
 
+@proceedings{PROC2017IJCCI,
+ editor = a_sabourin_christophe # { and } # a_merelo_guervos_juan_julian # { and } # a_oreilly_una_may # { and } # a_madani_kurosh # { and } # a_warwick_kevin,
+ booktitle = {Proceedings of the 9th International Joint Conference on Computational Intelligence {(IJCCI'17)}, } # nov # {~1-3, 2017, } # l_portugal_funchal,
+ date = {2017},
+ isbn = {978-989-758-274-5},
+ xdata = {c_ijcci}
+}
+
 @article{PS2009FYOSASOM,
  author = a_potts_chris_n # { and } # a_strusevich_vitaly_a,
  title = {Fifty Years of Scheduling: {A} Survey of Milestones},
@@ -1404,6 +1449,17 @@ @article{PS2009FYOSASOM
  urldate = {2023-12-06}
 }
 
+@article{RLMC2022MNOGAASCCH,
+ author = a_ravber_miha # { and } # a_liu_shih_hsi # { and } # a_mernik_marjan # { and } # a_crepinsek_matej,
+ titl = {Maximum Number of Generations as a Stopping Criterion Considered Harmful},
+ xdata = {j_asoc},
+ volume = {128},
+ number = {109478},
+ date = {2022-10},
+ doi = {10.1016/J.ASOC.2022.109478},
+ url = {https://www.researchgate.net/publication/362640980}
+}
+
 @article{S1961TBWPAPA,
  author = {Leon Steinberg},
  title = {The Backboard Wiring Problem: A Placement Algorithm},

notation/optimization.sty

@@ -26,7 +26,7 @@ A search operator $\@searchOp:\random\times\searchSpace^k\mapsto\searchSpace$ wi
 \newSymbol{obspel}{\ensuremath{z}}{z}{%
 an objective value returned by the objective function~\objf, see \cref{def:objectiveValue}}%
 \newSymbol{goalF}{\ensuremath{z_g}}{zg}{%
-a goal objective value~$z_g$ is a pre-defined limit at which the optimization process can stop, i.e., if a solution~\solspel\ with $\objfOf{\solspel}\leq z_g$ is discovered, the optimization process can terminate and \shouldTerminate\ becomes \codeil{True}.}%
+a goal objective value~$z_g$ is a pre-defined limit at which the optimization process can stop, i.e., if a solution~\solspel\ with $\objfOf{\solspel}\leq z_g$ is discovered, the optimization process can terminate and \shouldTerminate\ becomes \codeil{True}, see \cref{sec:termination:defs}.}%
 %
 \DeclareMathOperator{\@shouldTerminate}{shouldTerminate}%
 \newSymbol{shouldTerminate}{\ensuremath{\@shouldTerminate}}{shouldTerminate}{%

text/main/structure/termination/termination.tex

@@ -1,37 +1,45 @@
-\hsection{The Termination Criterion and the Problem of Measuring Time}%
+\hsection{The Termination Criterion}%
 \label{sec:terminationCriterion}%
 %
 We have seen that the search spaces for even small instances of the \gls{JSSP} can already be quite large.
 We simply cannot enumerate all points in them, as it would take too long.
-This raises the question:
+While optimization algorithms usually do something much cleverer than simply checking all possible solutions, we already discovered the profoundly simply question:
 \inQuotes{If we cannot look at all possible solutions, how can we find the global optimum?}
-We may also ask:
+We then went a step further and asked:
 \inQuotes{If we cannot look at all possible solutions, how can we know whether a given candidate solution is the global optimum or not?}
 In some optimization scenarios, we can have good lower bounds~\lowerBoundOf{\objf} of the objective function~\objf\ and if we find a solution~$\globalOptimumOf{\solspel}\in\solutionSpace$ with $\objfOf{\globalOptimumOf{\solspel}}\leq\lowerBoundOf{\objf}$, then we know that we have solved the problem.
 However, usually the answer to both question is simply:
 \emph{We cannot.}
 
-Usually, we don't know if the best solution~\bestSoFarOf{\solspel} we know so far is the global optimum or not.
+Usually, we simply don't know if the best solution~\bestSoFarOf{\solspel} we know so far is the global optimum or not.
 This leads us to another problem:
-If we do not know whether we found the best-possible solution or not, how do we know if we can stop the optimization process or should continue trying to solve the problem?
+If we do not know whether we found the best-possible solution or not, how do we know if we can stop the optimization process (whatever that might be) or should continue trying to solve the problem?
 
-There are two very simple answers to this question:
-Either when the time is up or when we found a reasonably-good solution.
-And there are some less simple answers, such as \inQuotes{when we were not able to improve upon~\bestSoFarOf{\solspel} for a while.}
+At first glance, there are two very simple answers to this question:
+We stop either \inQuotes{when the time is up} or \inQuotes{when we found a reasonably-good solution.}
+Additionally, there are many more complicated answers~\cite{GCJ2017TCIEAAS}, such as \inQuotes{when we were not able to improve upon~\bestSoFarOf{\solspel} for a while.}%
 %
 \hsection{Definitions}%
+\label{sec:termination:defs}%
 %
-\begin{definition}[Termintion Criterion]%
+\begin{definition}[Termination Criterion]%
 \label{def:terminationCriterion}%
 The \emph{termination criterion}~$\shouldTerminate:\mathSpace{S}\mapsto\{\codeil{False}, \codeil{True}\}$ is a function of the state~\mathSpace{S} of the optimization process which becomes \codeil{True} if the optimization process should stop and remains \codeil{False} as long as it can continue.%
 \end{definition}%
 %
 Into such a termination criterion we can embed any combination of time or solution quality limits.
 We could, for instance, define a goal objective value~\goalF\ good enough so that we can stop the optimization procedure as soon as a candidate solution~$\solspel\in\solutionSpace$ has been discovered with $\objfOf{\solspel}\leq\goalF$, i.e., which is at least as good as the goal.
 We could set $\goalF=\lowerBoundOf{\objf}$ if we have a reasonably good lower bound.
-We could also set to some acceptable quality limit resulting from our practical application scenario, 
+We could also set to some acceptable quality limit resulting from our practical application scenario.
 
-Alternatively -- or in addition -- we may define a maximum amount of time the user is willing to wait for an answer, i.e., a computational budget after which we simply need to stop.%
+Alternatively -- or in addition -- we may define a maximum amount of time the user is willing to wait for an answer, i.e., a computational budget after which we simply need to stop.
+
+Most optimization processes iteratively try to find better solutions.
+We could stop when, for a certain amount of time, no better solution was discovered.
+While the current-best solution may not be \inQuotes{reasonably good,} continuing to try finding better solutions may be pointless anyway when the last improvement was made one minute after we started the optimization process {\dots} and, by now, that is ten hours ago{\dots}
+
+Some algorithms maintain sets of solutions and iteratively work on all the solutions in the sets.
+A maximum number of such iterations was used as stopping criterion in some older works, but this is known to be a bad practice and, thus, strongly discouraged~\cite{RLMC2022MNOGAASCCH}.%
 %
 \endhsection%
 %
@@ -74,7 +82,15 @@
 \endhsection%
 %
 \hsection{Summary}%
-TODO
+In this section, we raised the question when an optimization should stop.
+This is an important question, since we already know that we usually do not know whether we found the optimal solution or not.
+Sometimes, termination criteria arise from our practical scenario, e.g., our application defines a time limit and then, that is when we will stop.
+Sometimes, we have a lot of time available and could let the optimization process go on for quite a bit.
+However, even then, we do not want to waste computational time when it is already clear that our algorithm cannot make any improvement anymore.
+Finally, we may have a goal quality threshold~\goalF\ and after reaching it, we can stop.
+Sometimes we have a combination of the above.
+Either way, we have to stop eventually.%
 \endhsection%
 \endhsection%
 %
+