## Operations research: applications and algorithms |

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12-2

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

/(jc,,x2>- ••>*») be a

12-2

**CONVEX**AND CONCAVE FUNCTIONS**Convex**and concave functions playan 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 ...Page 621

\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

\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, then ...Page 624

GROUP A On the given set 5, determine whether each

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; ...

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; ...

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### Contents

BASIC LINEAR ALGEBRA | 9 |

l | 42 |

INTRODUCTION TO LINEAR PROGRAMMING | 51 |

Copyright | |

24 other sections not shown

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### 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