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Convex Set of Points

Convex Set of Points. A set of points is convex if it has the following property: Consider all possible pairs of points in the set and consider the line segment connecting any such pair. All such line segments must lie entirely within the set. Convex –vs- Nonconvex.

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Convex Set of Points

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  1. Convex Set of Points • A set of points is convex if it has the following property: Consider all possible pairs of points in the set and consider the line segment connecting any such pair. All such line segments must lie entirely within the set.

  2. Convex –vs- Nonconvex

  3. Highly Nonlinear Problems Highly Nonlinear Problems General Convex Problems Quadratic Problems Linear Problems

  4. Quadratic Programming • Linear convex constraints • Objective is a quadratic function: • Ai xi2 +  Bijxixj +  Cixi + D • Portfolio structuring uses quadratic programming models • Unique optimal solution exists that can be found using Solver’s generalized reduced gradient (GRG) algorithm

  5. General convex models: • Feasible region is a general convex set • Objective may or may not be a general convex function • Produces a unique solution • Highly nonlinear models: • Convexity does not exist • Local and global optimal solutions are possible

  6. Risk Solver Platform • Convexity Tester: • Optimize, Analyze Without Solving or the X-Checkbox icon on the Model tab in the task pane • Results: NSP – Non-smooth problem which is non-convex, NLP - Non-linear smooth problem which may or may not be convex • Model Type – NLP Convex in the Model Diagnosis area of the task pane implies that the local optimal solution is a global optimal solution • Use RSPE’s Guided Mode to interpret your messages and types of nonlinearity in model

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