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▲❛r❣❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ❇❡♥❞❡rs✬ s❡❛r❝❤ ❙t❡♣❤❡♥ ❏✳ ▼❛❤❡r ▲❛♥❝❛st❡r ❯♥✐✈❡rs✐t② ▼❛♥❛❣❡♠❡♥t ❙❝❤♦♦❧✱ s✳♠❛❤❡r✸❅❧❛♥❝❛st❡r✳❛❝✳✉❦ ✽t❤ ▼❛r❝❤ ✷✵✶✽ ✶ ✴ ✷✸

❙tr✉❝t✉r❡❞ ♠✐①❡❞ ✐♥t❡❣❡r ♣r♦❣r❛♠♠✐♥❣ ❇❛s✐❝ ✐❞❡❛✿ ▼✐♥✐♠✐s❡ ❛ ❧✐♥❡❛r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♦✈❡r ❛ s❡t ♦❢ s♦❧✉t✐♦♥s s❛t✐s❢②✐♥❣ ❛ str✉❝t✉r❡❞ s❡t ♦❢ ❧✐♥❡❛r ❝♦♥str❛✐♥ts✳ c ⊤ x + d ⊤ y , ♠✐♥ s✉❜❥❡❝t t♦ Ax ≥ b , Bx + Dy ≥ g , x ∈ Z p ✶ + × R n ✶ − p ✶ , + y ∈ Z p ✷ + × R n ✷ − p ✷ . + ✷ ✴ ✷✸

❙♦❧✈✐♥❣ str✉❝t✉r❡❞ ♠✐①❡❞ ✐♥t❡❣❡r ♣r♦❣r❛♠s ◮ ❙t❛t❡✲♦❢✲t❤❡✲❛rt ❣❡♥❡r❛❧ ♣✉r♣♦s❡ s♦❧✈❡rs ◮ ❈P▲❊❳✱ ●✉r♦❜✐✱ ❳♣r❡ss✱ ❙❈■P ✱ ✳ ✳ ✳ bound clique shift convex domcol proj convert close tworow int to cuts trivial bin bnd aggre cgmip dual gation disjunc agg tive dual sym comp eccuts infer least metry ence inf zero full half sym strong most Presolver dual Separator break infer gauge inf strong distri stuff bution ing cg multi gate rapid extract aggr impl learn gomory Branch uct free ing implied spars cloud restart bounds ify redvub intto node dfs mcf reopt binary odd impl cycle int allfull ics hybrid qpkkt obj shift& simple strong estim ref pscost prop rounding rootsol sync relps random Node diving cost trivial selector esti negation repair shifting mate rounding subnlp trivial or quadratic rins twoopt Impli rens dfs Tree trysol cations reoptsols orbisack soc breadth bfs pseudo first default under vbounds boolean orbitope setppc pscost random cover diving rounding Cutpool veclen logicor nonlinear sos1 sos2 zero diving objective proximity oneopt Conflict Dialog super zi linking indicator rounding sym ofins linear resack SCIP Constraint octane Primal actcons diving Handler Heuristic knapsack varbound objpscost diving alns bound integral xor Relax Event nlp diving abs indicator power mutation coef clique Expr. diving cum Interpr. bivariate multi mpec disjunc- ulative start and tion conjunc- cardin crossover tion ality NLP complete sol CppAD locks bound distrib count worhp Pricer lpface disjunc · · · ution sols linesearch comp tion diving onents diving feaspump dins cpx ipopt filter local intdiving guided frac sqp branching wbo bnd diving diving dualval int spx2 grb zpl shifting fixand rlp sol indicator gins infer LP ccg cip nlobbt genv obbt bounds spx1 msk ppm cnf dualfix orbital fixing clp none pip Reader diff Propa xprs qso pbm v gator fix probing bounds osil opb gms sync pseudo fzn obj mps root redcost redcost mst lp ✸ ✴ ✷✸

❙♦❧✈✐♥❣ str✉❝t✉r❡❞ ♠✐①❡❞ ✐♥t❡❣❡r ♣r♦❣r❛♠s ◮ ❙t❛t❡✲♦❢✲t❤❡✲❛rt ❣❡♥❡r❛❧ ♣✉r♣♦s❡ s♦❧✈❡rs ◮ ❈P▲❊❳✱ ●✉r♦❜✐✱ ❳♣r❡ss✱ ❙❈■P ✱ ✳ ✳ ✳ ◮ ❉❡❝♦♠♣♦s✐t✐♦♥ t❡❝❤♥✐q✉❡s ◮ ❈♦❧✉♠♥ ❣❡♥❡r❛t✐♦♥✱ ▲❛❣r❛♥❣✐❛♥ r❡❧❛①❛t✐♦♥✴❞❡❝♦♠♣♦s✐t✐♦♥✱ ❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✳ ✳ ✳ bound clique shift convex domcol proj convert close tworow int to cuts trivial bin bnd aggre cgmip dual gation disjunc agg tive dual sym comp eccuts infer least metry ence inf zero full half sym strong most Presolver dual Separator break infer gauge inf strong distri stuff bution ing cg multi gate rapid extract aggr impl learn gomory Branch uct free ing implied spars cloud restart bounds ify redvub intto node dfs mcf reopt binary odd impl cycle int allfull ics hybrid qpkkt obj shift& simple strong estim ref pscost prop rounding rootsol sync relps random Node diving cost trivial selector esti negation repair shifting mate rounding subnlp trivial or quadratic rins twoopt Impli rens dfs Tree trysol cations reoptsols orbisack soc breadth bfs pseudo first default under vbounds boolean orbitope setppc pscost random cover diving rounding Cutpool veclen logicor nonlinear sos1 sos2 zero diving objective proximity oneopt Conflict Dialog super zi linking indicator rounding sym ofins linear resack SCIP Constraint octane Primal actcons diving Handler Heuristic knapsack varbound objpscost diving alns bound integral xor Relax Event nlp diving abs indicator power mutation coef clique Expr. diving cum Interpr. bivariate multi mpec disjunc- ulative start and tion conjunc- cardin crossover tion ality NLP complete sol CppAD locks bound distrib count worhp Pricer lpface disjunc · · · ution sols linesearch comp tion diving onents diving feaspump dins cpx ipopt filter local intdiving guided frac sqp branching wbo bnd diving diving dualval int spx2 grb zpl shifting fixand rlp sol indicator gins infer LP ccg cip nlobbt genv obbt bounds spx1 msk ppm cnf dualfix orbital fixing clp none pip Reader diff Propa xprs qso pbm v gator fix probing bounds osil opb gms sync pseudo fzn obj mps root redcost redcost mst lp ✸ ✴ ✷✸

❙♦❧✈✐♥❣ ❧❛r❣❡ s❝❛❧❡ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s ❆✐♠✿ ❊♠❜❡❞ ❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤✐♥ ❛ st❛t❡✲♦❢✲t❤❡✲❛rt s♦❧✈❡r t♦ ♣r♦✈✐❞❡ ❡✛❡❝t✐✈❡ t♦♦❧s t♦ ❡♠♣❧♦② ❞❡❝♦♠♣♦s✐t✐♦♥ t♦ s♦❧✈❡ ❧❛r❣❡✲s❝❛❧❡ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s✳ ✹ ✴ ✷✸

❙♦❧✈✐♥❣ ❧❛r❣❡ s❝❛❧❡ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s ❆✐♠✿ ❊♠❜❡❞ ❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤✐♥ ❛ st❛t❡✲♦❢✲t❤❡✲❛rt s♦❧✈❡r t♦ ♣r♦✈✐❞❡ ❡✛❡❝t✐✈❡ t♦♦❧s t♦ ❡♠♣❧♦② ❞❡❝♦♠♣♦s✐t✐♦♥ t♦ s♦❧✈❡ ❧❛r❣❡✲s❝❛❧❡ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s✳ ◮ ❉❡✈❡❧♦♣ ❛ ❣❡♥❡r❛❧ ❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ ❢r❛♠❡✇♦r❦ ◮ ❍❛r♥❡ss t❤❡ ❝❛♣❛❜✐❧✐t✐❡s ♦❢ st❛t❡✲♦❢✲t❤❡✲❛rt ▼■P s♦❧✈❡rs ✇❤❡♥ ✉s✐♥❣ ❞❡❝♦♠♣♦s✐t✐♦♥ t❡❝❤♥✐q✉❡s ◮ ■♥t❡❣r❛t✐♦♥ ♦❢ ❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❧❛r❣❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ s❡❛r❝❤ ❤❡✉r✐st✐❝s✳ ✹ ✴ ✷✸

❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ ❖r✐❣✐♥❛❧ ♣r♦❜❧❡♠ c ⊤ x + d ⊤ y , ♠✐♥ s✉❜❥❡❝t t♦ Ax ≥ b , Bx + Dy ≥ g , x ∈ Z p ✶ + × R n ✶ − p ✶ , + y ∈ R n ✷ + . ✺ ✴ ✷✸

❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ c ⊤ x + f ( x ) , ♠✐♥ s✉❜❥❡❝t t♦ Ax ≥ b , x ∈ Z p ✶ + × R n ✶ − p ✶ . + ✇❤❡r❡ { d ⊤ y | Bx + Dy ≥ g } f ( x ) = ♠✐♥ y ∈ R n ✷ + ✺ ✴ ✷✸

❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ ▼❛st❡r ♣r♦❜❧❡♠ ❙✉❜♣r♦❜❧❡♠ c ⊤ x + ϕ, d ⊤ y , ♠✐♥ z (ˆ x ) = ♠✐♥ s✉❜❥❡❝t t♦ Ax ≥ b , s✉❜❥❡❝t t♦ Dy ≥ g − B ˆ x , ϕ ≥ u ⊤ y ∈ R n ✷ − p ✷ ω ( g − Bx ) ∀ ω ∈ O , . + ✵ ≥ u ⊤ ω ( g − Bx ) ∀ ω ∈ F , ϕ ∈ R + , x ∈ Z p ✶ + × R n ✶ − p ✶ . + ✻ ✴ ✷✸

❇❡♥❞❡rs✬ ❞❡❝♦♠♣♦s✐t✐♦♥ ▼❛st❡r ♣r♦❜❧❡♠ ❙✉❜♣r♦❜❧❡♠ c ⊤ x + ϕ, d ⊤ y , ♠✐♥ z (ˆ x ) = ♠✐♥ s✉❜❥❡❝t t♦ Ax ≥ b , s✉❜❥❡❝t t♦ Dy ≥ g − B ˆ x , ϕ ≥ u ⊤ y ∈ Z p ✷ + × R n ✷ − p ✷ ω ( g − Bx ) ∀ ω ∈ O , . + ✵ ≥ u ⊤ ω ( g − Bx ) ∀ ω ∈ F , ④♥♦✲❣♦♦❞✴✐♥t❡❣❡r ❝✉ts⑥ , ϕ ∈ R + , x ∈ Z p ✶ + × R n ✶ − p ✶ . + ✻ ✴ ✷✸

❙t❛♥❞❛r❞ ❇❡♥❞❡rs✬ ✐♠♣❧❡♠❡♥t❛t✐♦♥ Start Solve master problem Solve subproblems z (ˆ x ) > ϕ Yes - add cut No Stop ◮ ❊❛s② t♦ ✉♥❞❡rst❛♥❞ ❛♥❞ s✐♠♣❧❡ t♦ ✐♠♣❧❡♠❡♥t✳ ◮ ◆♦t ❛❧✇❛②s ❡✛❡❝t✐✈❡✱ ❧❛r❣❡ ♦✈❡r❤❡❛❞ ✐♥ r❡♣❡❛t❡❞❧② s♦❧✈✐♥❣ ♠❛st❡r ♣r♦❜❧❡♠✳ ✼ ✴ ✷✸