The support function of a nonempty set is . It is convex and positively homogeneous, and for a closed convex set its convex conjugate is the indicator function of .
For a nonempty closed convex set , the finite-valued support function has subgradients exactly at the maximizing points of . This follows from Fenchel–Young inequality and the conjugacy between the support function and the indicator functional of a constraint set. For compact , the exposed maximizing set is nonempty.
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.