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Description of large scale optimization package, Version 2.1.0 |
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############################################################## |
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Patrick Heimbach, MIT/EAPS, 02-Mar-2000 |
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reference: |
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######### |
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J.C. Gilbert & C. Lemarechal |
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Some numerical experiments with variable-storage quasi-Newton algorithms |
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Mathematical Programming 45 (1989), pp. 407-435 |
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flow chart |
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########## |
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lsopt_top |
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|---- check arguments |
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|---- CALL INSTORE |
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| | |
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| |---- determine whether OPWARMI available: |
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| * if no: cold start: create OPWARMI |
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| * if yes: warm start: read from OPWARMI |
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| create or open OPWARMD |
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| |
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|---- check consistency between OPWARMI and model parameters |
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| |
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|---- >>> if COLD start: <<< |
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| | first simulation with f.g. xx_0; output: first ff_0, gg_0 |
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| | set first preconditioner value xdiff_0 to 1 |
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| | store xx(0), gg(0), xdiff(0) to OPWARMD (first 3 entries) |
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| | |
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| >>> else: WARM start: <<< |
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| read xx(i), gg(i) from OPWARMD (first 2 entries) |
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| for first warm start after cold start, i=0 |
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|---- /// if ITMAX > 0: perform optimization (increment loop index i) |
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| )---- save current values of gg(i-1) -> gold(i-1), ff -> fold(i-1) |
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| (---- CALL LSUPDXX |
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| ) | |
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| ( |---- >>> if jmax=0 <<< |
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| ) | | first optimization after cold start: |
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| ( | | preconditioner estimated via ff_0 - ff_(first guess) |
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| ) | | dd(i-1) = -gg(i-1)*preco |
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| ) | >>> if jmax > 0 <<< |
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| ( | dd(i-1) = -gg(i-1) |
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| ) | CALL HESSUPD |
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| ( | | |
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| ) | |---- dd(i-1) modified via Hessian approx. |
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| ( | |
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| ) |---- >>> if <dd,gg> >= 0 <<< |
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| ( | ifail = 4 |
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| ) | |
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| ( |---- compute step size: tact(i-1) |
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| ) |---- compute update: xdiff(i) = xx(i-1) + tact(i-1)*dd(i-1) |
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| )---- >>> if ifail = 4 <<< |
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| ( goto 1000 |
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| (---- CALL OPTLINE / LSLINE |
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| ( |---- /// loop over simulations |
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| ) ( |
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| ( )---- CALL SIMUL |
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| ) ( | |
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| ( ) |---- input: xdiff(i) |
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| ) ( |---- output: ff(i), gg(i) |
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| ( ) |---- >>> if ONLINE <<< |
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| ) ( runs model and adjoint |
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| ( ) >>> if OFFLINE <<< |
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| ) ( reads those values from file |
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| ( ) |
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| ) (---- 1st Wolfe test: |
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| ( ) ff(i) <= tact*xpara1*<gg(i-1),dd(i-1)> |
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| ) ( |
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| ( )---- 2nd Wolfe test: |
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| ) ( <gg(i),dd(i-1)> >= xpara2*<gg(i-1),dd(i-1)> |
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| ( ) |
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| ) (---- >>> if 1st and 2nd Wolfe tests ok <<< |
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| ( ) | 320: update xx: xx(i) = xdiff(i) |
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| ) ( | |
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| ( ) >>> else if 1st Wolfe test not ok <<< |
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| ) ( | 500: INTERpolate new tact: |
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| ( ) | barr*tact < tact < (1-barr)*tact |
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| ) ( | CALL CUBIC |
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| ( ) | |
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| ) ( >>> else if 2nd Wolfe test not ok <<< |
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| ( ) 350: EXTRApolate new tact: |
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| ) ( (1+barmin)*tact < tact < 10*tact |
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| ( ) CALL CUBIC |
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| ) ( |
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| ( )---- >>> if new tact > tmax <<< |
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| ) ( | ifail = 7 |
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| ( ) | |
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| ) (---- >>> if new tact < tmin OR tact*dd < machine precision <<< |
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| ( ) | ifail = 8 |
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| ) ( | |
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| ( )---- >>> else <<< |
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| ) ( update xdiff for new simulation |
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| ( ) |
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| ) \\\ if nfunc > 1: use inter-/extrapolated tact and xdiff |
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| ( for new simulation |
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| ) N.B.: new xx is thus not based on new gg, but |
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| ( rather on new step size tact |
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| ) |
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| (---- store new values xx(i), gg(i) to OPWARMD (first 2 entries) |
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| )---- >>> if ifail = 7,8,9 <<< |
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| ( goto 1000 |
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| ) |
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| (---- compute new pointers jmin, jmax to include latest values |
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| ) gg(i)-gg(i-1), xx(i)-xx(i-1) to Hessian matrix estimate |
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| (---- store gg(i)-gg(i-1), xx(i)-xx(i-1) to OPWARMD |
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| ) (entries 2*jmax+2, 2*jmax+3) |
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| )---- CALL DGSCALE |
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| ) |---- call dostore |
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| ) | |---- read preconditioner of previous iteration diag(i-1) |
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| ( | from OPWARMD (3rd entry) |
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| ) | |
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| ( |---- compute new preconditioner diag(i), based upon diag(i-1), |
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| ) | gg(i)-gg(i-1), xx(i)-xx(i-1) |
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| ) |---- call dostore |
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| ) |---- write new preconditioner diag(i) to OPWARMD (3rd entry) |
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|---- \\\ end of optimization iteration loop |
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|---- CALL OUTSTORE |
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| |---- store gnorm0, ff(i), current pointers jmin, jmax, iterabs to OPWARMI |
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|---- >>> if OFFLINE version <<< |
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| xx(i+1) needs to be computed as input for offline optimization |
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| |---- CALL LSUPDXX |
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| | |---- compute dd(i), tact(i) -> xdiff(i+1) = x(i) + tact(i)*dd(i) |
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| |---- CALL WRITE_CONTROL |
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| | |---- write xdiff(i+1) to special file for offline optim. |
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|---- print final information |
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O |
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Remarks: |
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####### |
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1. Difference between offline/online version |
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-------------------------------------------- |
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- Offline version: Every call to simul refers to a read procedure which |
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reads the result of an offline forward and adjoint run |
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Therefore, only one call to simul is allowed, |
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itmax = 0, for cold start |
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itmax = 1, for warm start |
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Also, at the end, x(i+1) needs to be computed and saved |
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to be available for the offline model and adjoint run |
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- Online version: Every call to simul refers to an execution of the forward and adjoint model. |
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Several iterations of optimization may thus be performed within |
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a single run of the main program (main_lsopt). |
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The following cases may occur: |
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- cold start only (no optimization) |
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- cold start & one or several iterations of optimization |
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- warm start from previous cold start with one or several iterations |
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- warm start from previous warm start with one or several iterations |
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In order to achieve minimum difference between the online and offline code |
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xdiff(i+1) is stored to file at the end of an (offline) iteration, |
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but recomputed identically at the beginning of the next iteration. |
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2. Number of iterations vs. number of simulations |
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------------------------------------------------- |
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- itmax: controls the max. number of iterations |
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- nfunc: controls the max. number of simulations within one iteration |
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Summary: From one iteration to the next the descent direction changes. |
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Within one iteration more than one forward and adjoint run may be performed. |
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The updated control used as input for these simulations uses the same |
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descent direction, but different step sizes. |
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In detail: |
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From one iteration to the next the descent direction dd changes using |
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the result for the adjoint vector gg of the previous iteration. |
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In lsline the updated control xdiff(i,1) = xx(i-1) + tact(i-1,1)*dd(i-1) serves as input for |
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a forward and adjoint model run yielding a new gg(i,1). |
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In general, the new solution passes the 1st and 2nd Wolfe tests |
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so xdiff(i,1) represents the solution sought: xx(i) = xdiff(i,1). |
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If one of the two tests fails, an inter- or extrapolation is invoked to determine |
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a new step size tact(i-1,2). |
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If more than one function call is permitted, the new step size is used together |
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with the "old" descent direction dd(i-1) (i.e. dd is not updated using the new gg(i)), |
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to compute a new xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1) that serves as input |
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in a new forward and adjoint run, yielding gg(i,2). |
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If now, both Wolfe tests are successfull, the updated solution is given by |
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xx(i) = xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1). |
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3. Double-usage of fields dd and xdiff |
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-------------------------------------- |
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In order to save memory both the fields dd and xdiff have a double usage. |
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- xdiff: in lsopt_top: used as x(i) - x(i-1) for Hessian update |
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in lsline: intermediate result for control update x = x + tact*dd |
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- dd : in lsopt_top, lsline: descent vector, dd = -gg & hessupd |
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in dgscale: intermediate result to compute new preconditioner |
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4. Notice for user of old code |
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------------------------------ |
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Three relevant changes needed to switch to new version: |
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(i): OPWARMI file: two variables added: |
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gnorm0 : norm of first (cold start) gradient |
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iabsiter: total number of iterations with respect to cold start |
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(ii): routine names that are referenced by main_lsopt.f |
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lsoptv1 -> lsopt_top |
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lsline1 -> lsline |
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(iii): parameter list of lsopt_top |
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logical loffline included |
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parameter file lsopt.par |
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######################## |
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The optimization is controlled by a set of parameters |
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provided through the standard input file lsopt.par, |
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which is generated within the job script. |
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NUPDATE : max. no. of update pairs (gg(i)-gg(i-1), xx(i)-xx(i-1)) |
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to be stored in OPWARMD to estimate Hessian |
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[pair of current iter. is stored in (2*jmax+2, 2*jmax+3) |
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jmax must be > 0 to access these entries] |
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Presently NUPDATE must be > 0 |
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(i.e. iteration without reference to previous |
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iterations through OPWARMD has not been tested) |
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EPSX : relative precision on xx bellow which xx should not be improved |
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EPSG : relative precision on gg below which optimization is considered successful |
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IPRINT : controls verbose (>=1) or non-verbose output |
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NUMITER : max. number of iterations of optimisation |
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NUMTER = 0: cold start only, no optimization |
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ITER_NUM : index of new restart file to be created (not necessarily = NUMITER!) |
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NFUNC : max. no. of simulations per iteration |
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(must be > 0) |
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is used if step size tact is inter-/extrapolated; |
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in this case, if NFUNC > 1, a new simulation is performed with |
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same gradient but "improved" step size |
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FMIN : first guess cost function value |
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(only used as long as first iteration not completed, |
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i.e. for jmax <= 0) |
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OPWARMI, OPWARMD files |
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###################### |
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Two files retain values of previous iterations which are |
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used in latest iteration to update Hessian. |
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OPWARMI: contains index settings and scalar variables |
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OPWARMD: contains vectors |
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Structure of OPWARMI file: |
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------------------------- |
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n, fc, isize, m, jmin, jmax, gnorm0, iabsiter |
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n = nn : no. of control variables |
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fc = ff : cost value of last iteration |
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isize : no. of bytes per record in OPWARMD |
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m = nupdate : max. no. of updates for Hessian |
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jmin, jmax : pointer indices for OPWARMD file (cf. below) |
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gnorm0 : norm of first (cold start) gradient gg |
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iabsiter : total number of iterations with respect to cold start |
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Structure of OPWARMD file: |
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------------------------- |
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entry |
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1 : xx(i) : control vector of latest iteration |
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2 : gg(i) : gradient of latest iteration |
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3 : xdiff(i), diag: preconditioning vector; (1,...,1) for cold start |
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--- |
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2*jmax+2 : gold = g(i) - g(i-1) for last update (jmax) |
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2*jmax+3 : xdiff = tact * d = xx(i) - xx(i-1) for last update (jmax) |
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if jmax = 0: cold start; no Hessian update used to compute dd |
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if jmax > nupdate, old positions are overwritten, starting |
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with position pair (4,5) |
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Example 1: jmin = 1, jmax = 3, mupd = 5 |
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1 2 3 | 4 5 6 7 8 9 empty empty |
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|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___| |
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0 | 1 2 3 |
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Example 2: jmin = 3, jmax = 7, mupd = 5 ---> jmax = 2 |
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1 2 3 | |
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|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___| |
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Error handling |
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############## |
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ifail | description |
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--------+---------------------------------------------------------- |
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< 0 | should not appear (flag indic in simul.F not used) |
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0 | normal mode during execution |
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1 | an input argument is wrong |
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2 | warm start file is corrupted |
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3 | the initial gradient is too small |
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4 | the search direction is not a descent one |
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5 | maximal number of iterations reached |
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6 | maximal number of simulations reached (handled passively) |
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7 | the linesearch failed |
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8 | the function could not be improved |
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9 | optline parameters wrong |
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10 | cold start, no optimization done |
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11 | convergence achieved within precision |
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