Interactive lab¶
Here you can get your hands on optimization. Each app opens in a page of its own, at full size: drag the points, move the sliders and watch sets, functions and optima change.
1. Convex sets¶
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Convex combination
Three points, three weights: see how convex combinations fill the triangle.
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Convexity test
Convex and non-convex sets: does the segment between two points stay in the set?
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Balls in the q-norm
Unit balls of norms: from the diamond to the circle to the square.
The chapter: Convex sets.
2. Functions of several variables¶
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:material-contour: Level sets and gradient
The level curves of a function of two variables and the gradient, always perpendicular.
The chapter: Functions of several variables.
3. Convex and concave functions¶
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The chord and the graph
The definition of convex function: does the graph lie below the chord?
The chapter: Convex and concave functions.
4. Quadratic functions¶
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The matrix Q
Quadratic functions: how the matrix \(Q\) determines the shape of level sets and convexity.
The chapter: Quadratic functions.
5. Optimization problems¶
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Optimum on an interval
Optimal, unbounded, bound not attained, infeasible: the outcomes of an optimization problem.
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Local and global optima
Neighborhoods, local and global minima of a function on a feasible set.
The chapter: Optimization problems.
6. Convex and concave optimization problems¶
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Descent to a local optimum
Where does a local method stop? In convex problems every local minimum is global.
The chapter: Convex and concave optimization problems.