KnowraConvex analysisConvex analysisConvex analysis studies convex sets and functions, especially their geometric structure, optimization, and duality.BriefConnectConvex set: A set is convex when every line segment between two of its points lies entirely within the set. Convex sets are the geometric objects whose intersections and boundaries underpin the subject.Convex optimization: Convex optimization minimizes a convex objective over a convex feasible set. It is the principal problem class where convex analysis supplies global guarantees.Linear programming: Linear programming optimizes a linear objective over a region defined by linear inequalities. It is a foundational convex problem with geometric and dual interpretations.Nonconvex optimization: Nonconvex optimization studies problems whose objectives or feasible sets are not convex. Without convexity, local minima may fail to be global and duality gaps can appear.Hermann Minkowski: Hermann Minkowski was a mathematician whose work shaped geometry of numbers and convex geometry. His geometric methods helped establish the modern study of convex bodies.Convex function: A function is convex when its value along any line segment is at most the linear interpolation of its endpoint values. Convex functions have tractable minima and support much of convex optimization.Lagrangian duality: Lagrangian duality constructs bounds and a dual optimization problem from constraints and a primal objective. Convexity often makes its bounds exact, linking primal solutions to certificates.Least squares: Least squares estimates parameters by minimizing the sum of squared residuals. Its convex quadratic objective makes it a standard model for convex optimization.Concave function: A concave function lies above the chord joining any two points on its graph. Concavity reverses the defining inequality for convex functions and often models maximization.Werner Fenchel: Werner Fenchel was a mathematician known for foundational work on convexity and duality. His work gave convex conjugacy and duality much of their modern form.Show all 28Linked from 9 pagesHahn–Banach theoremRelated: The theorem's separation form connects functional extension with convex geometry.Extended real number lineNarrower topic: Convex functions often take the value positive infinity to encode constraints.Hotelling's lemmaRelated: Convexity clarifies when profit derivatives exist and how they identify optimal choices.Karamata's inequalityNarrower topic: The inequality's proof and extensions draw on the structure of convex functions.Young's inequality for productsRelated: Convex analysis places the power inequality within a broad theory of conjugate functions.Levinson's inequalityNarrower topic: Levinson's inequality is one result in the broader theory of convex-function inequalities.Shapley–Folkman lemmaNarrower topic: The lemma became a core tool for reasoning about sums and convexification.Show all 9