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Quasiconcavity, Quasiconvexity

Quasiconcavity and Quasiconvexity . If we know the concavity or convexity of the objective function, no need to check the second-order condition . In constrained optimization, it is possible to dispense with the second-order condition if the surface or hypersurface has the appropriate type of confi

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Quasiconcavity, Quasiconvexity

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    1. Quasiconcavity, Quasiconvexity

    2. Quasiconcavity and Quasiconvexity If we know the concavity or convexity of the objective function, no need to check the second-order condition . In constrained optimization, it is possible to dispense with the second-order condition if the surface or hypersurface has the appropriate type of configuration.

    3. Quasiconcavity and Quasiconvexity The desired configuration is: quasiconcavity (rather than concavity) for a maximum, and quasiconvexity (rather than convexity) for a minimum. These are weaker conditions than concavity and convexity. These can also be strict or non-strict. This is only to be expected, since the second-order sufficient condition to be dispensed with is also weaker for the constrained optimization problem (d2z definite in sign only for those dxi satisfying dg = 0) than for the free one (d2z definite in sign for all dxi).This is only to be expected, since the second-order sufficient condition to be dispensed with is also weaker for the constrained optimization problem (d2z definite in sign only for those dxi satisfying dg = 0) than for the free one (d2z definite in sign for all dxi).

    10. Algebraic Definitions

    11. 3 Theorems Theorem I (negative of a function) If f(x) is quasiconcave (strictly quasiconcave), then --f(x) is quasiconvex (strictly quasiconvex). Theorem II (concavity versus quasiconcavity) Any concave (convex) function is quasiconcave (quasiconvex), but the converse is not true. Similarly, any strictly concave (strictly convex) function is strictly quasiconcave (strictly quasiconvex), but the converse is not true. Theorem III (linear function) If f(x) is a linear function, then it is quasiconcave as well as quasiconvex.

    12. Theorem I - Proof Theorem I follows from the fact that multiplying an inequality by -1 reverses the sense of inequality. Let f(x) be quasiconcave,? with f(v) > f(u). Then, f[? u + (1 - ?)v] > f (u). As far as the function -f(x) is concerned, however, we have (after multiplying the two inequalities through by -1) -f(u) > -f(v) and -f[? u + (1 - ?)v] < -f (u). Interpreting -f(u) as the height of point N, and -f(v) as the height of M, we see that the function - f(x) satisfies the condition for quasiconvexity.

    13. Theorem II - Proof

    14. Theorem III - Proof

    17. Example 1: Check z= x2 (x > 0) for quasiconcavity and quasiconvexity

    19. Example 2 Show that z = f(x, y) = xy (with x, y > 0) is quasiconcave

    20. Example 3

    21. Differentiable Functions

    22. Differentiable Functions

    23. Differentiable Functions

    24. Differentiable Functions

    27. Bordered Determinant vs. Bordered Hessian

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