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For this new LP, it s easy to come up with a starting vertex, namely, the one with z i = bi for all i and all other variables zero Therefore we can solve it by simplex, to obtain the optimum solution There are two cases If the optimum value of z 1 + + zm is zero, then all zi s obtained by simplex are zero, and hence from the optimum vertex of the new LP we get a starting feasible vertex of the original LP, just by ignoring the z i s We can at last start simplex! But what if the optimum objective turns out to be positive Let us think We tried to minimize the sum of the zi s, but simplex decided that it cannot be zero But this means that the original linear program is infeasible: it needs some nonzero z i s to become feasible This is how simplex discovers and reports that an LP is infeasible Degeneracy In the polyhedron of Figure 712 vertex B is degenerate Geometrically, this means that it is the intersection of more than n = 3 faces of the polyhedron (in this case, 2 , 3 , 4 , 5 ) Algebraically, it means that if we choose any one of four sets of three inequalities ({ 2 , 3 , 4 }, { 2 , 3 , 5 }, { 2 , 4 , 5 }, and { 3 , 4 , 5 }) and solve the corresponding system of three linear equations in three unknowns, we ll get the same solution in all four cases: (0, 300, 100) This is a serious problem: simplex may return a suboptimal degenerate vertex simply because all its neighbors are identical to it and thus have no better objective And if we modify simplex so that it detects degeneracy and continues to hop from vertex to vertex despite lack of any improvement in the cost, it may end up looping forever One way to x this is by a perturbation: change each b i by a tiny random amount to bi i This doesn t change the essence of the LP since the i s are tiny, but it has the effect of differentiating between the solutions of the linear systems To see why geometrically, imagine that the four planes 2 , 3 , 4 , 5 were jolted a little Wouldn t vertex B split into two vertices, very 214.

earlier in this chapter for the Finder window, as shown in boldface here:

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dbvacctmgr@aos>-- create the end-user access account dbvacctmgr@aos> CREATE USER ellen_enduser IDENTIFIED BY <password>; User created dbvacctmgr@aos>-- expire the password for the account so they are dbvacctmgr@aos>-- forced to change it to a password dbvacctmgr@aos>-- they can control at next login dbvacctmgr@aos>ALTER USER ellen_enduser PASSWORD EXPIRE; User altered dbvacctmgr@aos>CONNECT jean_oper_dba Enter password: Connected jean_oper_dba@aos>-- provide the ability to connect to the database GRANT CREATE SESSION TO ellen_enduser; Grant succeeded jean_oper_dba@aos>-- we have to login as the Sales History Security jean_oper_dba@aos>-- administrator to perform the grants jean_oper_dba@aos>-- of these end-user access roles jean_oper_dba@aos>CONNECT sam_sec_mgr Enter password: Connected sam_sec_mgr@aos>GRANT sh_rw_role_0101 TO ellen_enduser; Grant succeeded sam_sec_mgr@aos>GRANT sh_exec_role_0101 TO ellen_enduser; Grant succeeded sam_sec_mgr@aos>-- now connect as the end-user account sam_sec_mgr@aos>-- and see how it works sam_sec_mgr@aos>CONNECT ellen_enduser Enter password: ERROR: ORA-28001: the password has expired Changing password for ellen_enduser New password: Retype new password: Password changed Connected ellen_enduser@aos>-- the account can query the COSTS table ellen_enduser@aos>SELECT * FROM shcosts WHERE ROWNUM < 6; PROD_ID TIME_ID PROMO_ID CHANNEL_ID UNIT_COST UNIT_PRICE ---------- --------- ---------- ---------- ---------- ---------13 20-JAN-98 999 2 79314 120599 13 30-JAN-98 999 3 78303 123216 13 21-FEB-98 999 3 81307 123731 13 25-FEB-98 999 3 79869 123299 14 19-JAN-98 999 2 88645 110899

close to one another Unboundedness In some cases an LP is unbounded, in that its objective function can be made arbitrarily large (or small, if it s a minimization problem) If this is the case, simplex will discover it: in exploring the neighborhood of a vertex, it will notice that taking out an inequality and adding another leads to an underdetermined system of equations that has an in nity of solutions And in fact (this is an easy test) the space of solutions contains a whole line across which the objective can become larger and larger, all the way to In this case simplex halts and complains

tell application "TextEdit" make new document set the text of the front document to "Latest Report" & return & "Here is the latest news from the front" & return & "Sales have doubled!" tell the front document tell the first paragraph set the font to "Arial Bold" set the size to 18 end tell tell the second paragraph set the font to "Arial" set the size to 12 end tell end tell set the bounds of the front window to {800, 22, 1400, 822} end tell

5 rows selected ellen_enduser@aos>-- the account can delete the COSTS table ellen_enduser@aos>DELETE shcosts WHERE prod_id = 22; 1221 rows deleted ellen_enduser@aos>-- undo the transaction as it is only a test! ellen_enduser@aos>ROLLBACK; Rollback complete

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