A risk manager computing the worst-case P&L for a stress test over a bounded parameter region cannot rely on finding a critical point in the interior — the maximum loss might occur on the boundary of the region (the prescribed stress scenarios). The extreme value theorem guarantees that a continuous function on a closed bounded (compact) domain attains its global maximum and minimum. Finding them requires a two-phase search: interior critical points and all boundary points. On a polygon or a curved boundary, the boundary search reduces to a sequence of single-variable optimisation problems. This note develops the algorithm and applies it to a stress-testing example.
Learning objectives. By the end of this note you should be able to:
- state the extreme value theorem for functions of several variables;
- execute the boundary–critical-point algorithm to find global extrema on a compact domain;
- reduce a curved or polygonal boundary search to single-variable problems via parametrisation;
- identify when a constrained extremum occurs on the boundary versus the interior.
In this lesson
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- The Extreme Value Theorem
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- Where Extrema Can Occur
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- The Boundary–Critical-Point Algorithm
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- Rectangular Domain
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- Circular Boundary
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- Stress Testing Application
Before you start
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