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Outer approximation for quadratic facility location: solving different problem types in sequence Description A quadratic facility location problem is formulated and solved as an MIQP problem (quadfacloc.mos) and via an outer approximation algorithm that iterates over MIP and QP subproblems (quadfaclocoa.mos).
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quadfaclocoa.mos (!********************************************************************* Mosel NL examples ================= file quadfaclocoa.mos ````````````````````` Quadratic Facility Location model Model of a quadratic facility location problem solved by outer approximation (main problem: MIP, subproblem: QP). The goal is to assign facilities to customers to minimize cost. Original AMPL model by Sven Leyffer, see https://www.researchgate.net/publication/2827082_Solving_Mixed_Integer_Nonlinear_Programs_by_Outer_Approximation (c) 2018 Fair Isaac Corporation author: Z.Csizmadia, Sven Leyffer, Aug. 2018, rev. Apr.2021 *********************************************************************!) model QuadFacLocOA uses "mmxprs", "mmnl", "mmsystem" parameters ILIM = 100 end-parameters forward procedure printsol declarations Iter: integer end-declarations declarations I = 1..5 ! Set of facilities J = 1..10 ! Set of customers FacX,FacY: array(I) of real ! X/Y coordinates of facility location CustX,CustY: array(J) of real ! X/Y coordinates of customer location C: array(I) of real ! Fixed cost of opening facility at location Q: array(I,J) of real ! Quadr. cost of meeting demand j from facility i UBD,LBD: real ! Upper/lower bound on solution end-declarations ! **** Generate random data: ! **** Cost coefficients and coordinates of all facility + customer locations setrandseed(42) SCALE:=100 forall(i in I) do C(i) := SCALE*100*random FacX(i) := SCALE*random FacY(i) := SCALE*random end-do forall(j in J) do CustX(j) := SCALE*random CustY(j) := SCALE*random end-do forall(i in I, j in J) ! Formula used by Lee to create instances Q(i,j) := 50 * sqrt( (FacX(i) - CustX(j))*(FacX(i) - CustX(j)) + (FacY(i) - CustY(j))*(FacY(i) - CustY(j))) ! **** Optimization model formulation **** declarations eta: mpvar ! Linearization of quad. objective terms x: array(I,J) of mpvar ! Percentage of customer j served by facility j z: array(I) of mpvar ! =1, iff facility i built LastPoint: array(I) of real ! Latest discrete solution values from main LastMain: real ! Latest obj solution value for main problem LastX: array(I,J) of real ! Latest 'x' solution values from subproblem LastSub: real ! Latest obj solution value for subproblem end-declarations ! **** Main problem (MIP) declarations MainProb: mpproblem end-declarations with MainProb do declarations LinCost: linctr end-declarations eta is_free ! Linearized objective function LinCost:= sum(i in I) C(i)*z(i) ! After the initial iteration, the objective is changed to this form: ! LinCost:= sum(i in I) C(i)*z(i) + eta ! Variable upper bound (only serve customers from open facilities) forall(i in I, j in J) x(i,j) <= z(i) ! Meet all customers' demands forall(j in J) sum(i in I) x(i,j) = 1 forall(i in I) z(i) is_binary ! setparam("xprs_verbose",true) end-do ! **** Submodel (continuous QP) declarations SubNLP: mpproblem end-declarations with SubNLP do declarations QuadCost: nlctr Fixings: array(I) of linctr end-declarations ! Quadratic objective function QuadCost:= sum(i in I) C(i)*z(i) + sum(i in I, j in J) Q(i,j)*x(i,j)^2; ! Variable upper bound (only serve customers from open facilities) forall(i in I, j in J) x(i,j) <= z(i) ! Meet all customers' demands forall(j in J) sum(i in I) x(i,j) = 1 ! setparam("xprs_verbose",true) end-do ! **** Outer approximation algorithm UBD := 1E20; LBD := -1E20 Iter:= 0; starttime:=gettime while (LBD < UBD-(SCALE*1E-2) and Iter <= ILIM) do Iter += 1 with MainProb do minimize(LinCost) ! writeprob('MainProb_'+Iter,'l') if getprobstat<>XPRS_OPT then break; end-if forall(i in I) LastPoint(i) := getsol(z(i)) LastMain := getobjval end-do with SubNLP do forall(i in I) Fixings(i):= z(i) = LastPoint(i) ! Fix discrete variables minimize(QuadCost) ! writeprob('SubNLP_'+Iter,'l') forall(i in I, j in J) LastX(i,j) := getsol(x(i,j)) LastSub := getobjval end-do with MainProb do ! After the first iteration add the linearization term to the objective if Iter=1 then LinCost+= eta; end-if ! Linearization of quadratic part eta >= sum(i in I, j in J) Q(i,j)*LastX(i,j)^2 + ! Objective value sum(i in I, j in J) 2*Q(i,j)*LastX(i,j)*(x(i,j) - LastX(i,j)) ! Gradient end-do UBD := minlist( UBD, LastSub ) LBD := LastMain writeln(gettime-starttime, "sec Iter = ", Iter, ": LBD = ", LBD, " UBD = ", UBD) ! Uncomment this line to display intermediate solutions in detail: ! printsol end-do if Iter > ILIM then writeln("Iteration limit reached"); end-if printsol ! **** Display the solution **** procedure printsol write("Open Customer locations\nfacility") forall(j in J) write(formattext("%6d",j)) forall(i in I | LastPoint(i)>0) do write("\n ", textfmt(i,-8)) forall(j in J) write(formattext("%5.1f%%",LastX(i,j).sol*100)) end-do writeln end-procedure end-model | |||||||||||||

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