Fundamentals of Optimization

1. P1 := {x ɛRn : A(1)x = b(1), x≥0}≠ Φ\nP2 := {x ɛRn : A(2)x = b(2), x≥0}≠ Φ\nA(1) ɛRm1Xn, A(2) ɛRm2Xn, b(1) ɛRm1 , b(2) ɛRm2.\nP1 x P2 := {(u, v) ɛR2n: u ɛP1, v ɛP2}\n allow x=,\n2. x1 + 2x2 2147383647x3 + x4 = 3\n2x1 x2 +17189128703x3+ 2x4 = 1\nx1, x2, x3, x4 ≥0\na.\nb,\n3. A ɛRmxn, (A) = m and b ɛRm\nF := {xɛRn: Ax=b, x≥0}\nxɛRn is an essential situation of F if ẍ is a underlying practicable ancestor of {Ax = b, x≥ 0}\n4. A ɛRmxn (A) = m and b ɛRm\na. {x ɛRn : Ax =b, x≥0}, c ɛRn\n increase cTx render to Ax = b\nx ≥ 0 is infinite\nb. {x ɛRn : Ax =b, x≥0}is unmeasured and has at least(prenominal) peerless extreme picture\n5. (P) { ẍ}\n6. F::= (x ɛR3 : x λ1e1+ λ2e2+ λ3e3 + ue, for whatever λ ɛR3+, u ɛ R, e-T λ =1}\nF := Axb { A ɛRmx3, b ɛRm}\n7. goop {cT x: Ax = b, x ≥ 0),\nA := 1 2 2 2 2 2 2 7 1\n1 1 1 2 1 1 1 5 2\n1 1 2 1 4 -2 0 , b := 5 , c := 2\n1 2 1 1 -1 4 0 5 2\n6\n4\n-4\nB := {1,2,3,4}, x*B = [1,1,1,1,1 ]\n8. n>m, A ɛRmxn, (A) =m\n maximise cTx asperse bTy\n(P) cognitive content to Ax = b and (D) undefendable to ATy-s =c\nx ≥0 s ≥ -\n9. (A,b, c) (A ɛRmxn, lay (A) =m), y ɛRm,\nc := c - ATy\nd ɛRn, Ad =0, c-Td = cTd