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Lecture 5: Maxima and Minima of Convex Functions Advanced Mathematical Programming IE417 Lecture 5 Dr. Ted Ralphs IE417 Lecture 5 1 Reading for This Lecture ² Chapter 3, Sections 4-5 ² Chapter 4, Section 1 1 IE417 Lecture 5 2 Maxima and Minima of Convex Functions 2 IE417 Lecture 5 3 Minimizing a Convex Function Theorem 1. Let S be a nonempty convex set on Rn and let f : S ! R be ¤ convex on S. Suppose that x is a local optimal solution to minx2S f(x). ² Then x¤ is a global optimal solution. ² If either x¤ is a strict local optimum or f is strictly convex, then x¤ is the unique global optimal solution. 3 IE417 Lecture 5 4 Necessary and Sufficient Conditions Theorem 2. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. The point x¤ 2 S is an optimal solution to the problem T ¤ minx2S f(x) if and only if f has a subgradient » such that » (x ¡ x ) ¸ 0 8x 2 S. ² This implies that if S is open, then x¤ is an optimal solution if and only if there is a zero subgradient of f at x¤. 4 IE417 Lecture 5 5 Alternative Optima Theorem 3. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. If the point x¤ 2 S is an optimal solution to the problem minx2S f(x), then the set of alternative optima are characterized by the set S¤ = fx 2 S : rf(x¤)T (x ¡ x¤) · 0 and rf(x¤) = rf(x) Corollaries: 5 IE417 Lecture 5 6 Maximizing a Convex Function Theorem 4. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. If the point x¤ 2 S is an optimal solution to the problem T ¤ maxx2S f(x), then » (x ¡ x ) · 0 8x 2 S, where » is any subgradient of f. Theorem 5. Let S be a nonempty compact polyhedral set on Rn and let f : S ! R be convex on S. An optimal solution x¤ to the problem ¤ maxx2S f(x) exists where x is an extreme point of S. 6 IE417 Lecture 5 7 Importance of Convexity ² Allows easy approximation. ² Allows easy global optimization. ² Allows easy constrained optimization over polyhedral sets. 7 IE417 Lecture 5 8 Generalizing Convexity 8 IE417 Lecture 5 9 Quasiconvex Functions Definition 1. Let S be a nonempty convex set in Rn and let f : S ! R. The function f is said to be quasiconvex if, for each x1; x2 2 S, we have f[¸x1 + (1 ¡ ¸x2)] · maxff(x1); f(x2)g Theorem 6. Let S be a nonempty convex set in Rn and let f : S ! R. The function f is quasiconvex if and only if S® = fx 2 S : f(x) · ®g is convex for each ® 2 R. 9 IE417 Lecture 5 10 Strictly Quasiconvex Functions Definition 2. Let S be a nonempty convex set in Rn and let f : S ! R. The function f is said to be strictly quasiconvex if, for each x1; x2 2 S with f(x1) 6= f(x2), we have f[¸x1 + (1 ¡ ¸x2)] < maxff(x1); f(x2)g Theorem 7. Let S be a nonempty convex set in Rn and let f : S ! R be strictly quasiconvex. Consider the problem of minimizing f over S. If xˆ is a local optimal solution, then xˆ is a global optimal solution. ² Note that strict quasiconvexity does not imply quasiconvexity unless the function is lower semicontinuous. 10 IE417 Lecture 5 11 Strongly Quasiconvex Functions Definition 3. Let S be a nonempty convex set in Rn and let f : S ! R. The function f is said to be strictly quasiconvex if, for each x1; x2 2 S with x1 6= x2, we have f[¸x1 + (1 ¡ ¸x2)] < maxff(x1); f(x2)g Theorem 8. Let S be a nonempty convex set in Rn and let f : S ! R be strictly quasiconvex. Consider the problem of minimizing f over S. If xˆ is a local optimal solution, then xˆ is the unique global optimal solution. 11 IE417 Lecture 5 12 Pseudoconvex Functions Definition 4. Let S be a nonempty convex set on Rn and let f : S ! R be differentiable on S. The function f is said to be pseudoconvex if T 8x1; x2 2 S, with rf(x1) (x2 ¡ x1) ¸ 0, we have f(x2) ¸ f(x1). ² Strictly pseudoconvex can be defined analogously. ² Notice that all convex functions are pseudoconvex. ² Also, for a pseudoconvex function f, if rf(x¤) = 0, then x¤ is a local minimum. 12 IE417 Lecture 5 13 Local Convexity Definition 5. Let S be a nonempty convex set on Rn and let f : S ! R. Then the following are local properties of f. ² f is said to be convex at x¤ if f(¸x¤ + (1 ¡ ¸)x) · ¸f(x¤) + (1 ¡ ¸)f(x) for each ¸ 2 (0; 1) and x 2 S. ² f is said to be pseudoconvex at x¤ if rf(x¤)T (x ¡ x¤) ¸ 0 implies f(x) ¸ f(x¤). 13.
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