Convex optimization algorithms and complexity

  • Is convex optimization hard?

    In fact, solving a mixed integer convex problem is usually much harder than solving a nonconvex NLP to local optimality.
    For convex NLPs, it comes down to solving a local optimisation problem, however satisfying the optimality conditions is also NP-hard..

  • Is convex optimization hard?

    With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming..

  • What language is convex optimization?

    CVXPY is a domain-specific language for convex optimization embedded in Python.
    It allows the user to express convex optimization problems in a natural syntax that follows the math, rather than in the restrictive standard form required by solvers..

  • A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing.
    Linear functions are convex, so linear programming problems are convex problems.
  • Convex optimization has become an essential tool in machine learning because many real-world problems can be modeled as convex optimization problems.
    For example, in classification problems, the goal is to find the best hyperplane that separates the data points into different classes.
Convex optimization algorithms and complexity
Convex optimization algorithms and complexity

Smallest convex set containing a given set

In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it.
The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset.
For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

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