## Operations research: applications and algorithms |

### From inside the book

Results 1-3 of 37

Page 618

12-2 CONVEX AND CONCAVE FUNCTIONS Convex and concave functions play

an extremely important role in the study of nonlinear programming problems. Let

/(jc,,x2>- ••>*») be a function that is defined for all points (xux2,. . .,x„) in a convex

set S.+ DEFINITION A function /(^|,Jc2,. ..,x„) is a concave function on a convex

set S if for any x' e S and x" eS f(cx' + (1 - c)x-) 2: c/(x') + (1 - c)/(x") holds for 0 < c £

I . (3) Figure 4 A

...

12-2 CONVEX AND CONCAVE FUNCTIONS Convex and concave functions play

an extremely important role in the study of nonlinear programming problems. Let

/(jc,,x2>- ••>*») be a function that is defined for all points (xux2,. . .,x„) in a convex

set S.+ DEFINITION A function /(^|,Jc2,. ..,x„) is a concave function on a convex

set S if for any x' e S and x" eS f(cx' + (1 - c)x-) 2: c/(x') + (1 - c)/(x") holds for 0 < c £

I . (3) Figure 4 A

**Convex Function**From (2) and (3), we see that f(xl,x2,. . .,x„) is a...

Page 621

Similar reasoning can be used to prove Theorem 1' (See Problem 10 at the end

of this section). THEOREM 1' Consider NLP (1) and assume it is a minimization

problem. Suppose the feasible region 5 for NLP (1) is a convex set. \if(x) is

convex on 5, then any local minimum for NLP (1) is an optimal solution to this

NLP. Theorems 1 and 1 ' demonstrate that if we are maximizing a concave

function (or minimizing a

any local maximum ...

Similar reasoning can be used to prove Theorem 1' (See Problem 10 at the end

of this section). THEOREM 1' Consider NLP (1) and assume it is a minimization

problem. Suppose the feasible region 5 for NLP (1) is a convex set. \if(x) is

convex on 5, then any local minimum for NLP (1) is an optimal solution to this

NLP. Theorems 1 and 1 ' demonstrate that if we are maximizing a concave

function (or minimizing a

**convex function**) over a convex feasible region S, thenany local maximum ...

Page 624

GROUP A On the given set 5, determine whether each function is convex,

concave, or neither. 1. f(x) = x3; S = [0, oo) 2. f{x) = x5; 5 = R1 3. ftx) - i; S = (0, oo)

v4. /(x) = x" (0 5S a < 1); S = (0, oo) 5. f{x) = lnx;S = (0,oo) 6. /(x,,x2) = x\ + 3x,x2 +

xf; S = /?2 7./(x,,x2) = x2 + x22;S = *2 8. /(x,,x2) = — x2 — x,x2 — 2x1; S = R2 9.

For what values of a, b, and c will ax2 + 6x,x2 + cx2 be a

A concave function on R21 GROUP B 10. Prove Theorem 1 '. 11. Show that if /(x!,

x2,...,x„) ...

GROUP A On the given set 5, determine whether each function is convex,

concave, or neither. 1. f(x) = x3; S = [0, oo) 2. f{x) = x5; 5 = R1 3. ftx) - i; S = (0, oo)

v4. /(x) = x" (0 5S a < 1); S = (0, oo) 5. f{x) = lnx;S = (0,oo) 6. /(x,,x2) = x\ + 3x,x2 +

xf; S = /?2 7./(x,,x2) = x2 + x22;S = *2 8. /(x,,x2) = — x2 — x,x2 — 2x1; S = R2 9.

For what values of a, b, and c will ax2 + 6x,x2 + cx2 be a

**convex function**on R21A concave function on R21 GROUP B 10. Prove Theorem 1 '. 11. Show that if /(x!,

x2,...,x„) ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

BASIC LINEAR ALGEBRA | 9 |

l | 42 |

INTRODUCTION TO LINEAR PROGRAMMING | 51 |

Copyright | |

24 other sections not shown

### Other editions - View all

### Common terms and phrases

annual assume basic feasible solution basic variable basis remains optimal choose coefficient in row concave function constraint convex function convex set current basis remains Dakota problem decision maker decision variables decrease demand determine dual enter the basis example feasible region Figure following LP Formulate an LP Giapetto goal goal programming Golden Section Search holding cost increase integer programming investment labor LINDO output linear programming lottery LP relaxation matrix max problem maximize maximum maximum flow problem maxz method minimize minimum node nonbasic variable nonnegative objective function coefficient obtain optimal solution optimal tableau optimal z-value Powerco primal PROBLEMS GROUP profit programming problem purchased random variable reorder point requires right-hand side row player satisfy Section shadow price Shapley value Show shown in Table simplex algorithm Step subproblem Suppose Test Market Theorem utility function vector yields zero