Applying some special criteria ? Second-order conditions ? First-order conditions ? Reduction to a scalar function ? Showing that f is obtained through operations preserving convexity Convex Optimization 10 Lec
Network optimization: objective function and constraints have a special structure arising from a graph. In the following, the gradient ∇f(x) of f(x), x∈Rn, and the Hessian matrix {∇2f(x)}ij=∂2f(x)∂xi∂xj are denoted, respectively: ∇f(x)=[∂f∂x1⋮∂f∂xn]...
Convex Optimization Problems
The first variant is unconditional acceptance of each iterate. This is done in many popular SAO algorithms, such as CONLIN (Fleury and Braibant1986) and MMA (Svanberg1987), but also in many optimality criteria based optimization algorithms (see Groenwold and Etman2008a, and the references menti...
The main idea is to verify whether a quadratic function constructed from the Hessian matrix of U(⋅) (the matrix of 2nd partial derivatives) is a Sum-of-Squares (see Supplement). We can then formulate the problem of finding a convex polynomial underestimator of the sample points (ϕ(i...
The information processing objective of the method is to locate the extremum of a function. It does this by directly sampling the function using a pattern of three points. The points form the brackets on the search: the first and the last points are the current bounds of the search, and ...
energy. If we assume that𝟙, whereis a fixed positive constant and𝟙is the characteristic function of a subdomainsatisfying the constraint on the volume, then we find ourselves in the framework ofshape optimization[22,49,56]. In general, optimal shapes do not exists, and it is when the...
\({ {\mathcal{L}}}_{P}\)is also referred as a loss function, which usually describes the difference between the reconstructed intensityI(ϕ) and the object intensityIobj.I(ϕ) represents the reconstructed intensity as a function with respect to the POHϕ. Since the reconstructed intensi...
Let { x k } be a sequence produced by the DFSR1 and x ^ ∈ A such that V ( x ^ ) = 0 then { x k } converges to x ^ . Proof. Let the function V be Lipschitz continuous. First of all, we claim that lim k → ∞ inf ∥ V ( x k ) ∥ = 0 . (39) Assume (...
Each MINLP can be represented with a linear objective function vectorc. The objective and linear constraints inPare not required to be sparse; they may involve many or all variables inN, making them dense. The nonlinear functions, in many instances, are mainly supported by small subsets of the...