UNT 2 : NEQUALTES AND CONVEX SETS ' Structure 2. ntroducton Objectves, nequaltes and ther Graphs Convex Sets and ther Geometry Noton of Convex Sets Extreme Ponts of Convex Set Hyper Planes and Half Spaces 2.4 Summary
, : Lnear Programmng 2. NTRODUCTON h n a lnear programmng problem, we have constrants expressed n the form of lnear nequaltes. Therefore, to study lnear programmng, you must know the system of lnear nequaltes partcularly ther graphcal solutons. n ths unt we shall confne our dscusson to the graphcal solutons of nqualtes.,closely lnked wth the system of lnear nequaltes s the theory of convex sets. Ths theory has very mportant applcatons not only n lnear programmng but also n Economcs, Game Theory etc. ~ uto e these applcatons, a great deal of work has been done to develop the theory of Convex Sets. Thus, n ths unt, we dsucss the nequaltes and Convex Sets. n addton, we need the noton of Extreme Ponts, Hyper-plane and Half- Spaces. These notons wll be defned and explaned wth the help of some smple examples. Objectves After studyng ths unt, you should be able to draw the graph of the nequaltes and fnd ther solutons l defne a convex set and gve ts geometrcal descrpton defne an extreme pont, a hyper plane and a half space along wth ~ er graphcal llustratons. 2.2 NEQUALTES AND THER GRAPHS You are already famlar wth the concepts of lnes and ntersecton of lnes n a plane. You know that a general equaton of a lne s ax +by = c, where a, by c are real constants. t s also called a lnear equaton n two varables x and y. f we put y = 0, we get x = c/a, provded a ;t 0. Ths s the ntercept of the lne on X-axs. Smlarly on takng x = 0, we get - --!/ as the ntercept on Y-axs. By jonng the ponts (f. 0 and 0,, a # 0, b + 0. ) we an trace the lne on the graph paper..- For example consder the lqe 3x + 2y = 6. 40 -.-, -.-- --- ----- --- - - -
Draw ths lne on a graph paper as shown n Fgure. nequaltes and Convex - Sets Ths lne dvdes the plane nl.,,me sets or regons as shown n the fgure. These regons may be descrbed as follows: () The set of ponts (x~y) such that,.e. those ponts whch le on the lne. (Ch you gve a pont whch les o~ ths lne?) () The set of ponts (x, y) such that. \ 3x + 2 ~ ~ 6..For example the pont (0, 0) s such that 3.0 + 2.0 = 0 < 6 Agan the pont (, ) s such that,., 3. + 2. =5<6 The set of ponts (x, y) for whch 3x+2y<6, 4. ---
.J Basc Mathematcs and s called the half plane bounded by the lne Optmzaton r 3x+2y=6. () The set of ponts (x, y) such that 3x + 2y > 6. For example, consder the ponts You can see that (3,, (92. x = 0 and y = 0. 5(0) + 8(0) = 8 5 60. Note that the nequalty represents the set of ponts (x, y) whch ether le on the lne 3x + 2y = 6 or ' belong to the half plane 3x+2y<6. Smlarly, the nequalty 3x+2y26 ' represents the set of ponts (x, y) whch ether le on the lne 3x + 2y = 6 or belong to the half plane 3x + 2y > 6. Most of the nequaltes that we study n ths unt wll be of the form,! n general we can say that a lne ax + by = c $ : *, dvdes the XY plane nto three regons namely () the set of ponts (x, y) such that, ax + by = c, that s the lne tself; 42. 4d
() the set of ponts (x, y) such that ' Lnear Programmng.e. one of the the half planes bounded by the lne; () the set of ponts (x, y) such that ax +by > c, the other half plane bounded by the lne. Y For example, let us try to draw the graph of the nequalty Frst consder the lne. f we take y = 0, then x = 4. f x = 0, they y = 5/2. Therefore, we can trace the lne by jonng the ponts (4,O) and (0, 5/2). Let us now determne the locaton of the half-plane. For ths, we put Ths shows that x=oandy=o. 5(0) + 90) = 9 5 0.. -e s that half Fg. 2
Basc Mathc~atcs and Optmzaton plane n whch orgn does not le. Hence the shaded porton as shown n the fgure 2, represents' Let us consder now another example n whch we have more than one nequalty. At fst we wll be takng a smple case of three nequaltes. EXAMPLE : Graph the set of ponts (x, y) satsfyng the followng three nequaltes : and A = ((x, y) : 5x + 3y 2 5) B= ((x, y):x2o) SOLUTON : snce these nequaltes must be satsfed smultaneously, the desred set s the ntersecton of the three sets. The set A n B n C s the area whch s shaded and bounded by the lnet
You can note that the shaded regon conssts of the ponts lyng only n the ft quadrant due to the restrctons x2oandy20. nequaltes Convex ~ Qts ), ' EXAMPLE 2 : Graph the set of ponts whch satsfy the nequaltes x 2 and 3x + 4y 2 SOLUTON : You can draw the lne x =, whch s a vertcal lne tbugh the pont (,O). The lne s the lne jonng the ponts (4,O) and (0, 3). The set s shaded wth horzontal lnes and the set B= ((x,y)3x+4y<2) s the set wth vertcal shadng. Now the set of ponts whch satsfy both the nequaltes.e. the set A n B of ponts s the cross-hatched regon shown n the fgure 4.. \ Fg. 4...
nequaltes &Convex Sets The comer pont P s the ntersecton of lnes x = and 3x + 4y = 2.. Ths pont s Now we consder another example wth four nequaltes. EXAMPLE 3 : Graph the set of ponts whch satsfy the nequaltes SOLUTON : Draw the lnes 3x +2y = 8 andx.+2y = 0 \ by choosng two ponts on the lnes. Shade the correspondng regons Snce x20,y-20, therefore, the set of ponts wll be n the frst quadrant as shown n the fgure 5., Fg. 5 *
'The soluton set colssts of all the ponts of the quadrlateral regon OPQR, whch nclude the ponts of the boundry and the nteror regon. The vertces of the quadrlateral are nequaltes and Convex Sets where Q s the pont of ntersectol of the lnes 3x + 2y = 8 and x + 2y = 0. Let us now consder another example whch s slghtly dfferent. EXAMPLE 4 : Graph the followng nequaltes dentfy the polygon enclosed by these nequaltes. SOLUTON : You can verfy by drawng the lnes and shadng the conespondng regon, the polygo~l s gven by the Fgure 6. Fg. 6 C
Basc Mathematcs and The soluton set conssts of all the ponts of the polygon PQRS, whch Optmzato~~ nclude the ponts of the boundary as well as the nteror regon. The vertces of the polygon are Now tty the followng exercses. EXERCSE : Graph the set of ponts. whch satsfy the nequaltes EXERCSE 2 : Determne the set of ponts satsfed by each of the followng system of nequaltes. Graph that set and fnd ts corner ponts EXERCSE 3 : Graph the set of ponts whch satsfy the nequaltes Also fnd ts corner ponts. EXERCSE 4 : Graph the followng nequaltes dentfy the polygon enclosed by these nequaltes. Fnd the vertces of the ' polygon.
2.3 CONVEX SETS AND THER GEOMETRY nequaltes and Convex Sets n secton 2.2, we have dscussed how to fnd the set of ponts satsfyng the gven nequaltes. Ths set of ponts nay be a polyhedron or may not be a polyhedron. n ths sectol we shall study the behavour of these sets and a few other related notons, whch we need n Block 2, pa-tcularly n Unts 3 and 4. 2.3. Noton of Convex Sets Let x, and x, be any two ponts n the Eucldean space E n. Consder a lne passng through the ponts x,, x, (x,* x,) n E n defned as the set S = (x; x = hx, + (- h) x,, all real h ). B By gvng dfferent values to h we wll get the correspondng dfferent ponts on the lne. Suppose h s chosen such that 0 h. Then for h = 0, we get x = x, and for h =, we get x = 3. Thus, we get the v,uous ponts between x, and x,. The lne jonng these ponts correspondng to the values of h between 0 and, s often called a lne segment. n other words, the lne segment jonng the ponts x,, x, n E n correspondng to the values of h between 0 and s a set of ponts denoted by S, where S = (x:x=hxz+(-h)xl,05h<). Now we gve the defnton of a Convex Set.. DEFNTON : CONVEX SET A set S s sad to be Convex f for any two ponts x,, x, n the set, the lne segment jonng these ponts s also n the set. n other words, a set S s sad to be Convex f for any elements x,, x, E S, %+(~-?L)x,E SforO53t. For example, consder a trangle ABC and ts nteror as shown n Fgure 7,
Basc Mathematcs and Optmzaton Smlarly we can see that a square or a rectangle wth ts ncror s a Convex set as shaded n Fgure 8. Fg. 8 Agan, consder a set ((x, y) : x2 -t y7 ) whch s nfact, a crcle wlh ts nteror. Clearly t s a corlvex set. See Fgure 9, Fg. 9
Basc Mathematcs and A Opt~nzaton sets whch are convex and non-convex as shown n Fgure. Convex Convex Non Convex Fg. Non Convex Let us we state and prove the followng mportant theorem : THEOREM : f S, and S, are two Convex sets, then ther ntersecton ': s also a Convex set., PROOF : Suppose S, = S, n S, Let x,,'x, be any two ponts n S,. Then X,,X~E S,andx,,x,~ S,. Snce S, and S, are Convex, therefore, hx2+ ( -) x, ES, for 0 < h and hx, + ( - h) x, E S2 for 0 h &*, Hence 'hx2 + ( - h)x, E S, n S, for 0 5 h 5 J T Ths s hx, + ( - h)x, E S,O 5 h 5 52..
Thus S, s a convex set. nequaltes and Wth the help of followng examples you should verfy whether the gven sets are convex or not. EXAMPLE 5 : Show that the set n convex. SOLUTON : Suppose (x,, y,) E S, (x,, y,) E S are any two ponts. Then S wll be convex f h (x,, y2> + ( -J- (x,, Y,) E S O < h < l That s [Xx, + ( -h) x, hy, + ( -) y,l E S O s h r l Consder S 6h2 + 6 ( - h)2 + 2h ( - h) (3x,x2+ 2y, W) (Why?) Therefore 3[hx2 + ( - h)x,2 +2[3Ly2+( - h )~,]~ 2 6h2 + 6(- + 2h(l- h) (3xx2 + 2y,y2)....,( 2)
L Basc Matlematcs and Optmzaton Thus 3x,x2 + 2y,y2 6 From (2) and (3), we get Hence Ths shows that Therefore, S s a convex set. EXAMPLE 6 : Show that the set S((x, y) : xy 5, x 2 0, y 2 0) s not Convex. SOLUTON : n order to show that S s not convex we wll take two ponts n S and show that ther convex combnaton does not belong to S. Clearly, ( 3,) ~ and (-, 2) belong to S. Consder the combnaton of these ponts,.e. 2 S wll be convex f
Ths nequalty should hold for all values of h such that 0 h. But, f you take h = -, then you get 2 nequaltes and Convex fj, Sets /t Thus the nequalty s not satsfed for A = -. 2 Ths contradcton shows that S s not convex. EXERCSE 5 : Whch of the followng sets are convex? (a) s={(x, y) x ~ + ~ ~ > ~ } (b) s={(x, y) x2+3yz56} (c) S = {(X, y) x 5.2, y 5 4) Verfy your result by drawng the graph. 2.3.2 Extreme Ponts of a Convex Set ' n secton '(2.2), you studed the examples of convex sets n the fonn of polygons.the vertces of these polygons are mportant for us. We call such vertces as extreme ponts whch we defne now. DEFNTON 2 : EXTREME PONTS Let S be a Convex set, A pont x ES s an extreme pont of the convex set S f and only f there do not exst ponts x,, x, (x, + x,) n the set S such that a Let us look closely at ths defnton. Observe that the pont x = ( -h) x, + h x,, 0 < h < s a pont n between x, and x, (x, + x,).! ~ccordng to the above defrnton an extreme pont fals to satsfy ths property. For nstance consder a convex set S formed by a trangle ABC and ts nteror (See Fg. 2). f x s a pont nsde the trangle ABC, then t s possble to fnd pont x, qd x, n S such that J r f x s a boundery pont n S dfferent from ponts A, B and C, even then we can fnd ponts x, and x, n S such that 4" / l $! 4 '" ( x = hx, + ( -h) x,, 0 < h <. j 2 5 5
Basc Mathematcs and Optmzaton But for the ponts A, B and C ths s not possble. That s why the pont A, B and C are called the extreme ponts of S..,, Fg. 2 f we consder the crcle, then every pont on the crcumference of the crcle : s extreme pont. Why? Can you explan? Try t. Also, see the followng fgure.
,. ( 2.3.3 Hyper-Plane and Half Spaces We have seen n secton 2.2 that a lne dvdes the plane nto three parts. Consder an n-dmensonal plane whch dvdes the Eucldean Space E n nto.. three parts. Such a plane s called a Hyper plane. You know that c, x, + c, x, = d (c,, c,, d constants) s the equaton of a straght Lne n E'. Smlarly c, x, + c, x, + c, x, = d s the equaton of a plane ' j n Generalzng ths concept we many speak of the equaton of a hyperplane n E n. We defne the Hyper plane n the followng way : DEFNTON 3 : HYPER PLANE. nequaltes and / A hyper plane n E n s defned to be a set S of ponts : S={x En: clxl+cz~+...+ cnxn=d) S ={x En: cx=d) t where C = [el, $,..*, c,] ( A hyper plane cx = d n E" dvdes E n nto three mutually exclusve and exhaustve regons. These are denoted by the'sets : S,= (x: cx cd). S,= (x;cx=d) S, = {x : ox > d).'! The sets S, and S, are called Open half spaces, The sets S, = {x cx < d} and S, = {x ( cx 2 d} are called Closed half spaces. ' Note that S, n S, = S, whch s the hyper-plane cx = d.!'! NO& we wll show that hyper-planes, open half spaces and closed half spaces hre all convex sets. j THEOREM 2 : A hyperplane s a convex set --.. FROOF : n other words, we have to prove that the collecton '. S= (x~e":cx=d)! s a convex set.,'
Basc Mathematcs and Optmzaton f x,, x, are any two ponts on the hyper plane cx = d, then cx, = d andcx,=d The hyperplane wll be convex f the pont x=hx,+(l-h)x, for OL,<. les on the hyperplane. ndeed, we have cx = c [hx, t ( -A). x,] = kx, + ( - h) cx, =hdt(l-h)d=d : Ths shows that the hyperplane s a convex set, whch proves the theorem. THEOREM 3 : A closed half space s a convex set PROOF : Consder the closed half space Suppose x,, X, E S,, then cx, d, cx, 5 d. Consder x=hx,+(l-h)x, O<h<l Now cx= hcx,+(l-h)cx, $hd-t(l-h)d= d Ths shows that x= hx2+(-h)x, ES for 0 5 h Hence S s a Convex set. EXERCSE 6 : Show that S,, S, and S, are Convex sets. Now we wll gve few more defntons and theorems wthout proof whch wll be useful n Block 2. You may refer to the Book "Lnear Programmng" by G. Hadley for more detals. DEFNTON 4 : CONVEX COMBNATON k: 58 Let x,, x2,..., x, convex combnaton of ponts x,, xz,..., x,,, s defned as a pont be a fnte number of ponts n a Euldean space E n. A m x=xp,x,,p,>o =l,2,..., m =
where cp = We state a theorem wthout proof : nequaltes and Convex, Sets l j THEOREM 4 : The set sf all convex combnatons of a fnte number of ponts x,, x,,..., xn, n E n s a convex set that s, the set ( s convex. DEFWPTON 5 : CONVEX HULL Suppose A s a set whch s not convex. Then the smallest convex set whch contans A s called the Convex Hull of A. That s, the Convex Hull of a set A s the ntersecton of all Convex sets whch contan A. / For example, the Convex Hull of the set A = {(x, y) : x2 + yz = ) s the set klere S s Colvex and t contals A. Observe that S s the smallest convex set contanng A. n ths way we call say that the.convex -ull of the pont, on the crcunfere~ce of a crcle s the crcumference plus the nteror of the crcle. Ths s the slnallest convex set contanng the crcumference. ' We state another theorem wthout proof : THEOREM 5 : The convex hull of a fnte number of ponts x,, x,,..., x, n E n s the set of all convex combnaton of x,, $,..., x, L. That s, the convex hull of x,, x,,..., x, s the set The ~onvext ~ull of the fnte number of'ponts s called Convex Polyhedron
Basc Mathematcs and Optmzaton spannea ~y tnese pomts. ln Fgure 4, We navt: a LvvGx rolyneaton spanned by fve ponts. Fg. 4 f we consder the convex hull of three ponts n a plane t s a trangle. EXERCSE 7 : Suppose that X = (0, O), X2 = (2, O), X3 = (, ). Express the followng as convex lnear combnaton of the ponts x,, x, andx,: ',! 2.4 SUMMARY Recall that. ax + by S c and ax + by 2 c are nequaltes d 2. The lne ax + by = c dvdes the XY-plane nto three regons gven by the sets of... () ponts (x, y) such that ax + by = c * 60 '. >b. @,,,
() ponts (x, y) such that ax -t- by < c.e. one half of the plane bounded by the lne nequaltes & Convex Sets. () ponts(x,y)suchthatax-t-by>c,theotherhalfoftheplane., 3. A set S s convex f for any two ponts x,, x, n the set, the lne segment jonng these ponts s also n the set. 4. A pont x E S s an exkerne pont of the convex set S f there do not exst ponts x,, x, (x, # x,) n the set such that 5. The set [x E E n : cx = dl s a hyper plane n E n. The hyperplane cx = d dvdes E n nto three sets r' S,, S, are called Open half spaces. 6. Open half spaces, Closed half spaces and hyperplanes are Convex sets. 7. The Convex Hull of a set A s the smallest Convex set contanng A. The ( Convex Hull of a fnte number of ponts s the set of all Convex.combnaton of these ponts.
Basc Mathematcs and Optmzaton 2.5 ANSWERS/WNTS/SOLUT]ONS 4 6 0 4 El) Comer ponts are (0'4)'(-, 3 -), 3 (- 3 '-, 3 (0, 3) E2) a) Corner ponts are (0, -2)' (2, 0) and (-2, 2). - 5 4 25 2' 2 b) Comer ponts are (- ), (T, T) E3) Comer ponts are and (2' 0) E4) Vertces are E5) a) Not convex b) Convex C) Convex