qabel.gms : Linear Quadratic Control Problem

**Description**

This is a QCP formulation of the original ABEL model. Note that this model is convex and should be very easy to solve. The Linear Quadratic Riccati Equations are solved as a QCP. Nonlinear Programming Problem instead of the usual Matrix QMax can be easily extended to 1000, that allows to pass larger QPs to the solver.

**Reference**

- Kendrick, D, Caution and Probing in a Macroeconomic Model. Journal of Economic Dynamics and Control 4, 1 (1982), 149-170.

**Small Model of Type :** QCP

**Category :** GAMS Model library

**Main file :** qabel.gms

$Title Linear Quadratic Control Problem as QCP (QABEL,SEQ=293) $Ontext This is a QCP formulation of the original ABEL model. Note that this model is convex and should be very easy to solve. The Linear Quadratic Riccati Equations are solved as a QCP. Nonlinear Programming Problem instead of the usual Matrix QMax can be easily extended to 1000, that allows to pass larger QPs to the solver. Kendrick, D, Caution and Probing in a Macroeconomic Model. Journal of Economic Dynamics and Control 4, 2 (1982). $Offtext $if not set qmax $set qmax 75 Sets n states / consumpt, invest / m controls / gov-expend, money / k quarters / q1*q%qmax% / ku(k) control horizon ki(k) initial period kt(k) terminal period ; Alias (n,np), (m,mp) ; ku(k) = ord(k) < card(k); ki(k) = ord(k) = 1; kt(k) = not ku(k); Table a(n,np) state vector matrix consumpt invest consumpt .914 -.016 invest .097 .424 Table b(n,m) control vector matrix gov-expend money consumpt .305 .424 invest -.101 1.459 Table wk(n,np) penalty matrix for states - input consumpt invest consumpt .0625 invest 1 Table lambda(m,mp) penalty matrix for controls gov-expend money gov-expend 1 money .444 Parameter c(n) constant term / consumpt -59.4, invest -184.7 / xinit(n) initial value / consumpt 387.9, invest 85.3 / uinit(m) initial controls / gov-expend 110.5, money 147.1 / xtilde(n,k) desired path for x utilde(m,k) desired path for u w(n,np,k) penalty matrix on states ; w(n,np,ku) = wk(n,np); w(n,np,kt) = 100*wk(n,np); xtilde(n,k) = xinit(n)*power(1.0075,ord(k)-1); utilde(m,k) = uinit(m)*power(1.0075,ord(k)-1); * Display w, xtilde, utilde; Variables x(n,k) state variable u(m,k) control variable j criterion Equations criterion criterion definition stateq(n,k) state equation ; criterion.. j =e= .5*sum((k,n,np), (x(n,k)-xtilde(n,k))*w(n,np,k)*(x(np,k)-xtilde(np,k))) + .5*sum((ku,m,mp),(u(m,ku)-utilde(m,ku))*lambda(m,mp)*(u(mp,ku)-utilde(mp,ku))); stateq(n,k+1).. x(n,k+1) =e= sum(np, a(n,np)*x(np,k)) + sum(m, b(n,m)*u(m,k)) + c(n); Model abel /all/; x.l(n,k) = xinit(n); u.l(m,k) = uinit(m); x.fx(n,ki) = xinit(n); option limcol=0,limrow=0, solprint=off; Solve abel minimizing j using qcp; Display x.l, u.l;