The support function of a nonempty subset of a finite-dimensional real inner-product space isIt gives the largest value of a linear objective in direction , allowing when the set is unbounded in that direction. As a supremum of linear functions, it is convex and lower semicontinuous; it satisfies for and .
For the Euclidean unit ball, the Cauchy-Schwarz inequality gives , attained by when . For the probability simplex, the weighted-average bound gives , attained at a maximizing coordinate vector. Thus many familiar convex objectives are support functions.
Let be the indicator functional of a constraint set, equal to zero on and outside it. The convex conjugate definition gives . If is closed and convex, the Fenchel-Moreau theorem then gives . This uses the extended-real constraint indicator, rather than the zero-one indicator function used in measure theory.[ref]
The support-function subgradients as exposed faces identifies the maximizing vectors, linking sensitivity of the support function to the geometry of .
Let be a nonempty closed convex set in finite dimensions. At any where its support function is finite,Indeed for the indicator functional of a constraint set. The equality case of the Fenchel–Young inequality says exactly when , which is the displayed maximizing condition.
If is compact, the maximum is attained and this set is nonempty. For an unbounded closed convex set, a finite supremum need not be attained. Take , the epigraph of an exponential function, and . Then , approached as , but no point of has . The subdifferential at that direction is therefore empty. The support function must be considered at its finite-valued directions, rather than assumed finite everywhere for every closed convex set.
If , the support function is infinite along a kernel line. Otherwise linear programming duality gives the displayed minimum, attained because the corresponding linear programs are feasible with finite values. Splitting into nonnegative parts yields the equivalent minimum of . Thus worst-case revenue over a polyhedral uncertainty set is , with value off the row space.
For , linear programming duality giveswhere is the positive part of a real-valued function. To see equality directly, order the coordinates and choose for , or for . The threshold expression then equals the sum of the largest components. Introducing the variables turns this convex piecewise-linear objective into a linear program.
For a real symmetric matrix with eigenvalues , the sum of its largest eigenvalues is the support function of the fantope:Here is the Loewner order. In an orthonormal eigenbasis for , the objective depends only on the diagonal entries of , which lie in the capped simplex. Thus the identity reduces to the sum of the largest components of the eigenvalues.
For a real symmetric matrix and ,Indeed is a rank- orthogonal projection matrix in the fantope, and projecting onto the eigenvectors of the largest eigenvalues attains the support function maximum. Equivalently, the largest matrix trace of a compression to a -dimensional vector subspace is the sum of the largest eigenvalues.
For a finite-dimensional Hermitian operator with decreasing eigenvalues and integer , maximizing over positive contractions with trace gives the sum of its leading eigenvalues. In an orthonormal eigenbasis, the diagonal entries of lie in and sum to . Moving their weight to the largest eigenvalues can only increase the objective. An orthogonal projection onto their eigenvectors attains the maximum.
The sum of the largest eigenvalues has the semidefinite program representationFor every feasible in the fantope, positive semidefinite trace nonnegativity gives . To attain equality, use an orthonormal eigenbasis of and choose in that basis, with the same threshold choice as in the threshold formula for the sum of the largest components. This argument also covers , without requiring strict feasibility of the maximization program.
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The term "support function" can refer to different concepts depending on the context in which it is used. Here are a few common interpretations: 1. **Business Context**: In a business or organizational setting, support functions are departments or activities that assist the core operations of the business. Examples include human resources, IT support, customer service, and finance. These functions do not directly contribute to the production of goods or services but provide essential services that enable the core functions to operate smoothly.