Mathematics Operations Research Syllabus – Update 1

Mathematical Programming DefinitionsDefinitions Fundamental Theorem of Mathematical ProgrammingFundamental Theorem of Mathematical Programming –Mathematical Statement –Geometric Interpretation –Examples

Mathematical Programming  General Mathematical Programming Model  OPTIMIZEfxx S    X is an N - Dimensional Vector of DESICION VARIABLES  f is an OBJECTIVE FUNCTION  ()() {} S=: gx 0,= 1... n,hx= 0, = 1... m E Xand n i j ij  is a CONSTRAINT SET or FEASIBLE REGION in N- Dimensional Euclidean Space  ()() gx 0, hx= and i j i j  are the CONSTRAINT EQUATIONS that Define the Constraint Set  A Feasible Solution Vector, x S * , Which Optimizes the Objective Function, f, is the OPTIMUM FEASIBLE SOLUTION or Simply the OPTIMUM

DEFINITIONS  () ( ) {} N= S : d XX X, X   is an  - NEIGHBORHOOD ofX S   X A S  is a LIMIT POINT of A if ()     A N> 0 XXXX    A S  is CLOSED if it Contains Each of its Limit Points  A S  is BOUNDED if ()   > 0 A N X    A S  is COMPACT if it is Both Closed and Bounded EXAMPLES:1)X  0UNBOUNDED 2)0  X < 1NOT CLOSED 3)0  X  10COMPACT

Fundamental Theorem of Mathematical Programming Weierstrass Theorem Mathematical Statement A FUNCTION () f C X  DEFINED ON A COMPACT SET, S, HAS AN OPTIMUM, x * S  PROOF (MINIMUM CASE) () () {} Z 1 XZX fS= E : S = f    2) Every Compact Set of Real Numbers Contains its Greatest Lower Bound (GLB) ( ) () 1) i.e., Has A Compact Image on the Set of Real Numbers, 1 fS E X is Compact and X 1 S f C f : S E   () () 3) Z * GLB X * X * Z * = fS S f =    ( ) ( ) 4) ZXX * Z * X * is the Minimum x x* S, = f f = S      

A FUNCTION () f C X  DEFINED ON A COMPACT SET, S, HAS AN OPTIMUM, x * S  Feasible Region, S f(S) x x* z* Compact Image, f(S) Fundamental Theorem of Mathematical Programming Weierstrass Theorem Geometric Interpretation

 {} MAX x 2 x S : x 0    has no Solution Because S is unbounded and Therefore not Compact  {} MAX 10* x x S : 0 x< 1    has no Solution Because S is not closed And Therefore Not Compact  {} MAX xx S : 0< x 1 3    has a Solution at x = 1 Even Though S is not Compact S The Last Example Shows that the Conditions of the Weierstrass Theorem are SUFFICIENT but not NECESSARY Fundamental Theorem of Mathematical Programming Weierstrass Theorem Examples