UNCONSTRAINED OPTIMISATION MODULE CONVENOR: GANCARLO IANULARDO UNCONSTRAINED OPTIMISATION • Maximum and Minimum values • First and second order conditions CONDITIONS FOR A MAXIMUM OF A FUNCTION z f( x, y ) Conditions for a maximum of a function • z f( x, y ) Look at fig 1. The surface has a maximum at P. Consider the iso−y contour APB. In panel (b): z > 0 to left of P, z = 0 at P, and z < 0 to right of P. x x x Therefore in panel (c) the graph of z is downward sloping, x 2 2 so z is negative. So at P we have z = 0 and z< 0. x x 2 x 2 • Look at fig 2. By exactly the same reasoning as above, we conclude that at P we have z = 0 and 2z < 0. y y 2 FIGURE 15.1(A): EXAMINING AN ISO-Y SECTION TO FIND NECESSARY CONDITIONS FOR A MAXIMUM OF Z = F (X, Y) Surface has a maximum at P FIGURE 1(B & C): FINDING NECESSARY CONDITIONS FOR A MAXIMUM OF Z = F (X, Y) FIGURE 2(A): EXAMINING AN ISO-X SECTION TO FIND NECESSARY CONDITIONS FOR A MAXIMUM OF Z = F (X, Y) Surface has a maximum at P FIGURE 2 (B & C): FINDING NECESSARY CONDITIONS FOR A MAXIMUM OF Z = F (X, Y) Conditions for a minimum of a function Conditions for a minimum of a function z f( x, y ) • Look at fig 3, where there is a minimum at R. When we look at the iso−y contour ERF, we see that: z z 0 to left of R, z at R, and x 0 to right of R. 0 x x Therefore in panel (c) the graph of z is upward sloping, x 2 2 z z > 0. z So is positive. So at R we have = 0 and x 2 x 2 x • Look at fig 4. By exactly the same reasoning as above, 2 z z > 0. we conclude that at R we have y = 0 and y 2 FIGURE 3(A): EXAMINING AN ISO-Y SECTION TO FIND NECESSARY CONDITIONS FOR A MINIMUM OF Z = F Surface has a (X, Y) minimum at R FIGURE 3(B & C): FINDING NECESSARY CONDITIONS FOR A MINIMUM OF Z = F (X, Y) FIGURE 4(A): EXAMINING AN ISO-X SECTION TO FIND NECESSARY CONDITIONS FOR A MINIMUM OF Z = F (X, Y) Surface has a minimum at R FIGURE 4(B & C): FINDING NECESSARY CONDITIONS FOR A MINIMUM OF Z = F (X, Y) Collecting results: (rules 1 and 2) P in fig 1: MAX: z z 0 ; x y R in fig 3: MIN: z z 0 ; x y 2z 0 ; 2z 0 x 2 y 2 2z 0 ; 2z 0 x 2 y 2 Rules 1 and 2 are necessary but not sufficient for a max/min. This is because they refer only to variation in x or y separately. Saddle points Look at fig 6. From S, variation in only x or only y reduces z, but some variations in x and y together increase z. A point such as S is called a saddle point. FIGURE 6: HOW SEPARATE AND SIMULTANEOUS VARIATIONS IN X AND Y CAN HAVE DIFFERENT EFFECTS ON Z From S, any variation in x or y separately reduces z, but if x and y increase or decrease together, z may increase. SADDLE POINT To preclude a saddle point, we need in addition to rules 1 and 2: 2z 2z 2z 2z x 2 y 2 xy y x (rule 3) More on saddle points A point where z z 0 x y 2z 2z 2z 2z and (rule 3) x 2 y 2 xy y x definitely IS NOT a saddle point. A point where z z 0 x y and 2 z 2z 2 z 2 z x 2 y 2 xy y x reverse of rule 3) definitely IS a saddle point. (rule 4) (Note: case where 2z 2z 2z 2z x 2 y 2 xy y x remains open) (the A SPECIAL CASE Fig 7 shows one particular type of SP worth noting. Point S is a MIN for variation in x alone, so therefore z 0 and 2z 0 x x 2 Point S is a MAX for variation in y alone, so therefore z 0 and y 2z 0 y 2 Point S is also a saddle point, 2z 2z 2z 2z (rule 4) x 2 y 2 xy y x Key feature is that direct 2nd derivatives have opposite signs Renshaw: Maths for Economics, 4th edition FIGURE 7: A SADDLE POINT AT S THAT IS A MAXIMUM FOR VARIATION IN Y ALONE AND A MINIMUM FOR VARIATION IN X ALONE