When u is in the class of quasi convex functions, the L p norm of the length function I θ of the isoclines has minimizers with isoclines straight lines; the same occurs for other functionals on u depending on k, j. For a strictly regular quasi convex function isoclines may have ...
More details on the behavior of a convex function of two variables on the triangle and outside the triangle can be found in [9, Theorem 3.2]. Triangle cones have a prominent part in these considerations. The integral analogy of the concept of convex combination is the concept of barycenter....
IfMathML, q > 1, is a convex function on the co-ordinates onΔ, then the following inequality holds; MathML (2.2) where C is in the proof of Theorem 7. Proof. From Lemma 1, we have MathML By applying the well-known Hölder inequality for double integrals, then one has MathML...
Question 1: Convex, Concave, Quasi-convex, and Quasi-concave Functions Solutions for Question 1 Question 2: Perspective of a Function Solution for Question 2 Question 3: Operations that Preserve Convexity Solution for Question 3 Question 4: Conjugate Function Solution for Question 4 ...
Then, by taking\ln\left( \cdot \right) from objective function and both side of the first constraint, we have: \begin{array}{lll} \displaystyle{\min_{\bm{\widetilde{p}},t}} & \displaystyle{-\ln{t}} & \displaystyle{} \\ \displaystyle{\mathrm{subject\,\,to}} & \displaystyle{\...
of partial derivativesis nota subgradient of convex function. For nonsmooth nonconvex functions, the possibility of computing a single subgradient needs a serious mathematical justification [17]. On the other hand, if we have an access to a program for computing the value of our function, then ...
Further, if there exists α > 0 such that f(γx,y(t)) is an α-strongly convex function of t, for all x, y∈ A, then f is called α-strongly convex. For differentiable functions, it is possible to write down first-order and second-order characterizations of convexity (Udriste, ...
Step 3: If the shape of the epigraph is convex, then it is a convex function! How do we know if a function is convex? The definition with the epigraph is simple to understand, but with functions with several variables it is kind of hard to visualize. So we need to study the function...
The concept of a convex function was first introduced to elementary calculus when discussing the necessary conditions for a minimum or maximum value of a differentiable function. The convex function was later recognized as an active area of study by [20]. In modern studies, a convex function is...
The directional derivative f′(x0,p) is a positively homogeneous convex function of p. Theorem 1.27 asserts that x*∈∂f(x0) if and only if 〈x0,x*〉−f(x0)=f*(x*). This simple fact will be used in the next investigations. Also, if f is a closed proper convex function, ...