************************************************************************ * * * Test problems for NonSmooth UNConstrained minimization * * * TNSUNC includes the following subroutines * * S STARTX Initiation of variables. * S FUNC Computation of the value and the subgradient * of the objective function. * * * Napsu Haarala (2003) * * Haarala M., Miettinen K. and Mäkelä M.M.: New Limited Memory * Bundle Method for Large-Scale Nonsmooth Optimization, Optimization * Methods and Software, Vol. 19, No. 6, 2004, 673-692. * * ************************************************************************ * * * SUBROUTINE STARTX * * * * * Purpose * * * * Initiation of X. * * * * Calling sequence * * * CALL STARTX(N,X,NEXT) * * * * Parameters * * * II N Number of variables. * RO X(N) Vector of variables. * RI NEXT Problem number. * * * * Problems * * * 1. Generalization of MAXQ (convex). * 2. Generalization of MXHILB (convex). * 3. Chained LQ (convex). * 4. Chained CB3 (convex). * 5. Chained CB3 2 (convex). * 6. Number of active faces (nonconvex). * 7. Nonsmooth generalization of Brown function 2 (nonconvex). * 8. Chained Mifflin 2 (nonconvex). * 9. Chained crescent (nonconvex). * 10. Chained crescent 2 (nonconvex). * * * Napsu Haarala (2003) * * Haarala M., Miettinen K. and Mäkelä M.M.: New Limited Memory * Bundle Method for Large-Scale Nonsmooth Optimization, Optimization * Methods and Software, Vol. 19, No. 6, 2004, 673-692.. * SUBROUTINE STARTX(N,X,NEXT) * Scalar Arguments INTEGER N,NEXT * Array Arguments DOUBLE PRECISION X(*) * Local Arguments INTEGER I GOTO(10,20,30,40,40,60,70,80,90,90) NEXT PRINT*,'Error: Not such a problem.' NEXT=-1 RETURN * * Generalization of MAXQ (convex) * 10 CONTINUE DO 11 I=1,N/2 X(I) = DBLE(I) 11 CONTINUE DO 12 I=N/2+1,N X(I) = -DBLE(I) 12 CONTINUE RETURN * * Generalization of MXHILB (convex) * 20 CONTINUE DO 21 I=1,N X(I) = 1.0D+00 21 CONTINUE RETURN * * Chained LQ (convex) * 30 CONTINUE DO 31 I=1,N X(I)=-0.50D+00 31 CONTINUE RETURN * * Chained CB3 1 and 2 (convex) * 40 CONTINUE DO 41 I=1,N X(I)=2.0D+00 41 CONTINUE RETURN * * Number of active faces (nonconvex) * 60 CONTINUE DO 61 I=1,N X(I) = 1.0D+00 61 CONTINUE RETURN * * Nonsmooth generalization of Brown function 2 (nonconvex) * 70 CONTINUE DO 71 I=1,N IF(MOD(I,2) .EQ. 1) THEN X(I) = -1.0D+00 ELSE X(I) = 1.0D+00 ENDIF 71 CONTINUE RETURN * * Chained Mifflin 2 (nonconvex) * 80 CONTINUE DO 81 I=1,N X(I) = -1.0D+00 81 CONTINUE RETURN * * Chained crescent (nonconvex) * 90 CONTINUE DO 91 I=1,N IF(MOD(I,2) .EQ. 1) THEN X(I) = -1.50D+00 ELSE X(I) = 2.0D+00 ENDIF 91 CONTINUE RETURN END ************************************************************************ * * * SUBROUTINE FUNC * * * * * Purpose * * * * Computation of the value and the subgradient of the objective * function. * * * * Calling sequence * * * CALL FUNC(N,X,F,G,NEXT) * * * * Parameters * * * II N Number of variables. * RI X(N) Vector of variables. * RI NEXT Problem number. * RO F Value of the objective function. * RO G(N) Subgradient of the objective function. * * * * Problems * * * 1. Generalization of MAXQ (convex). * 2. Generalization of MXHILB (convex). * 3. Chained LQ (convex). * 4. Chained CB3 (convex). * 5. Chained CB3 2 (convex). * 6. Number of active faces (nonconvex). * 7. Nonsmooth generalization of Brown function 2 (nonconvex). * 8. Chained Mifflin 2 (nonconvex). * 9. Chained crescent (nonconvex). * 10. Chained crescent 2 (nonconvex). * * * Napsu Haarala (2003) * * Haarala M., Miettinen K. and Mäkelä M.M.: New Limited Memory * Bundle Method for Large-Scale Nonsmooth Optimization, Optimization * Methods and Software, Vol. 19, No. 6, 2004, 673-692.. * * SUBROUTINE FUNC(N,X,F,G,NEXT) * Scalar Arguments INTEGER N,NEXT DOUBLE PRECISION F * Array Arguments DOUBLE PRECISION G(*),X(*) * Local Arguments INTEGER I,J,HIT DOUBLE PRECISION Y,TEMP2,TEMP3,A,B,C,D,P,Q * Intrinsic Functions INTRINSIC DABS,DMAX1,SIGN,DLOG,DEXP,DCOS,DSIN GOTO(10,20,30,40,50,60,70,80,90,100) NEXT PRINT*,'Error: Not such a problem.' NEXT=-1 RETURN * * Generalization of MAXQ (convex) * 10 CONTINUE F=X(1)*X(1) G(1)=0.0D+00 HIT=1 DO 11 I=2,N Y=X(I)*X(I) IF (Y .GT. F) THEN F=Y HIT=I END IF G(I)=0.0D+00 11 CONTINUE G(HIT)=2*X(HIT) RETURN * * Generalization of MXHILB (convex) * 20 CONTINUE F = 0.0D+00 HIT=1 DO 21 J = 1,N F = F + X(J)/DBLE(J) 21 CONTINUE G(1)=SIGN(1.0D+00,F) F = DABS(F) DO 22 I = 2,N TEMP2 = 0.0D0 DO 23 J = 1,N TEMP2 = TEMP2 + X(J)/DBLE(I+J-1) 23 CONTINUE G(I)=SIGN(1.0D+00,TEMP2) TEMP2 = DABS(TEMP2) IF (TEMP2 .GT. F) THEN F=TEMP2 HIT=I END IF 22 CONTINUE TEMP3=G(HIT) DO 24 J = 1,N G(J) = TEMP3/DBLE(HIT+J-1) 24 CONTINUE RETURN * * Chained LQ (convex) * 30 CONTINUE F=0.0D+00 G(1)=0.0D+00 DO 31 I=1,N-1 G(I+1)=0.0D+00 A = -X(I)-X(I+1) B = -X(I)-X(I+1)+(X(I)*X(I)+X(I+1)*X(I+1)-1.0D+00) IF (A .GE. B) THEN F=F+A G(I)=G(I)-1.0D+00 G(I+1)=-1.0D+00 ELSE F=F+B G(I)=G(I)-1.0D+00+2.0D+00*X(I) G(I+1)=-1.0D+00+2.0D+00*X(I+1) ENDIF 31 CONTINUE RETURN * * Chained CB3 (convex) * 40 CONTINUE F=0.0D+00 G(1)=0.0D+00 DO 41 I=1,N-1 G(I+1)=0.0D+00 A=X(I)*X(I)*X(I)*X(I)+X(I+1)*X(I+1) B=(2.0D+00-X(I))*(2.0D+00-X(I))+ & (2.0D+00-X(I+1))*(2.0D+00-X(I+1)) C= 2.0D+00*DEXP(-X(I)+X(I+1)) Y=DMAX1(A,B) Y=DMAX1(Y,C) IF (Y .EQ. A) THEN G(I)=G(I)+4.0D+00*X(I)*X(I)*X(I) G(I+1)=2.0D+00*X(I+1) ELSE IF (Y .EQ. B) THEN G(I)=G(I)+2.0D+00*X(I)-4.0D+00 G(I+1)=2.0D+00*X(I+1)-4.0D+00 ELSE G(I)= G(I) - C G(I+1)= C END IF F=F+Y 41 CONTINUE RETURN * * Chained CB3 2 (convex) * 50 CONTINUE F=0.0D+00 G(1)=0.0D+00 A=0.0D+00 B=0.0D+00 C=0.0D+00 DO 51 I=1,N-1 G(I+1)=0.0D+00 A=A+X(I)*X(I)*X(I)*X(I)+X(I+1)*X(I+1) B=B+(2.0D+00-X(I))*(2.0D+00-X(I))+ & (2.0D+00-X(I+1))*(2.0D+00-X(I+1)) C=C+2.0D+00*DEXP(-X(I)+X(I+1)) 51 CONTINUE F=DMAX1(A,B) F=DMAX1(F,C) IF (F .EQ. A) THEN DO 53 I=1,N-1 G(I)=G(I)+4.0D+00*X(I)*X(I)*X(I) G(I+1)=2.0D+00*X(I+1) 53 CONTINUE ELSE IF (F .EQ. B) THEN DO 54 I=1,N-1 G(I)=G(I)+2.0D+00*X(I)-4.0D+00 G(I+1)=2.0D+00*X(I+1)-4.0D+00 54 CONTINUE ELSE DO 55 I=1,N-1 G(I)= G(I) - 2.0D+00*DEXP(-X(I)+X(I+1)) G(I+1)= 2.0D+00*DEXP(-X(I)+X(I+1)) 55 CONTINUE END IF RETURN * * Number of active faces (nonconvex) * 60 CONTINUE TEMP3=1.0D+00 Y=-X(1) G(1)= 0.0D+00 F=DLOG(DABS(X(1))+1.0D+00) HIT=1 TEMP2=F DO 62 I=2,N Y=Y - X(I) G(I)= 0.0D+00 F=DMAX1(F,DLOG(DABS(X(I))+1.0D+00)) IF(F .GT. TEMP2) THEN HIT=I TEMP2=F END IF 62 CONTINUE F=DMAX1(F,DLOG(DABS(Y)+1.0D+00)) IF(F .GT. TEMP2) THEN IF (Y.GE.0.0D+00) TEMP3=-1.0D+00 DO 63 I=1,N G(I)= TEMP3*(1.0D+00/(DABS(Y)+1.0D+00)) 63 CONTINUE ELSE IF (X(HIT).LT.0.0D+00) TEMP3=-1.0D+00 G(HIT)=TEMP3*(1.0D+00/(DABS(X(HIT))+1.0D+00)) END IF RETURN * * Nonsmooth generalization of Brown function 2 (nonconvex) * 70 CONTINUE F=0.0D+00 G(1)=0.0D+00 DO 71 I=1,N-1 A=DABS(X(I)) B=DABS(X(I+1)) C=X(I)*X(I)+1.0D+00 D=X(I+1)*X(I+1)+1.0D+00 F=F+B**C+A**D P=0.0D+00 Q=0.0D+00 IF (X(I).LT.0.0D+00) THEN IF (B .GT. P) P=DLOG(B) G(I)=G(I)-D*A**(D-1.0D+00)+2.0D+00*X(I)*P*B**C ELSE IF (B .GT. P) P=DLOG(B) G(I)=G(I)+D*A**(D-1.0D+00)+2.0D+00*X(I)*P*B**C ENDIF IF (X(I+1).EQ.0.0D+00) THEN G(I+1)=0.0D+00 ELSE IF (X(I+1).LT.0.0D+00) THEN IF (A .GT. Q) Q=DLOG(A) G(I+1)=-C*B**(C-1.0D+00)+2.0D+00*X(I+1)*Q*A**D ELSE IF (A .GT. Q) Q=DLOG(A) G(I+1)=C*B**(C-1.0D+00)+2.0D+00*X(I+1)*Q*A**D ENDIF 71 CONTINUE RETURN * * Chained mifflin 2 (nonconvex) * 80 CONTINUE F=0.0D+00 G(1)=0.0D+00 DO 81 I=1,N-1 Y = X(I)*X(I) + X(I+1)*X(I+1) - 1.0D0 F = F -X(I) + 2.0D+00*Y + 1.75D+00*DABS(Y) Y = SIGN(3.5D+00,Y) + 4.0D+00 G(I) = G(I) + Y*X(I) - 1.0D+00 G(I+1) = Y*X(I+1) 81 CONTINUE RETURN * * Chained crescent (nonconvex) * 90 CONTINUE TEMP2=0.0D+00 TEMP3=0.0D+00 DO 91 I=1,N-1 TEMP2 = TEMP2 + X(I)*X(I) + (X(I+1)-1.0D+00)*(X(I+1)-1.0D+00) & + X(I+1) - 1.0D+00 TEMP3 = TEMP3 - X(I)*X(I) - (X(I+1)-1.0D+00)*(X(I+1)-1.0D+00) & + X(I+1) + 1.0D+00 91 CONTINUE F = DMAX1(TEMP2,TEMP3) G(1)=0.0D+00 IF (TEMP2 .GE. TEMP3) THEN DO 92 I=1,N-1 G(I)=G(I)+2.0D+00*X(I) G(I+1)=2.0D+00*(X(I+1)-1.0D+00) + 1.0D+00 92 CONTINUE ELSE DO 93 I=1,N-1 G(I)=G(I)-2.0D+00*X(I) G(I+1)=-2.0D+00*(X(I+1)-1.0D+00) + 1.0D+00 93 CONTINUE END IF RETURN * * Chained crescent 2 (nonconvex) * 100 CONTINUE F=0.0D+00 G(1)=0.0D+00 DO 101 I=1,N-1 TEMP2 = X(I)*X(I) + (X(I+1)-1.0D+00)*(X(I+1)-1.0D+00) & + X(I+1) - 1.0D+00 TEMP3 = - X(I)*X(I) - (X(I+1)-1.0D+00)*(X(I+1)-1.0D+00) & + X(I+1) + 1.0D+00 IF (TEMP2 .GE. TEMP3) THEN F=F+TEMP2 G(I)=G(I)+2.0D+00*X(I) G(I+1)=2.0D+00*(X(I+1)-1.0D+00) + 1.0D+00 ELSE F=F+TEMP3 G(I)=G(I)-2.0D+00*X(I) G(I+1)=-2.0D+00*(X(I+1)-1.0D+00) + 1.0D+00 END IF 101 CONTINUE RETURN END