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Nonlinear Optimization
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
However, three points do suffice to bracket a minimum (or maximum). If a<b<c and f is continuous on [a,c] then we know from the Extreme Value Theorem that f achieves both its maximum and its minimum value over [a,c]. If f(b)<f(a) and f(b)<f(c) then the maximum may be achieved at one of the endpoints but the minimum cannot be and so there must be a local minimum in the interior of the interval. We have bracketed a minimum of the function. This is the basis of the exhaustive search method for minimization (see Sec. 1.1): Take small steps in a given direction until a bracket is found, then repeat with a finer step size over that bracket.
A conservative expected travel time approach for traffic information dissemination under uncertainty
Published in Transportmetrica B: Transport Dynamics, 2023
Ruiya Chen, Xiangdong Xu, Anthony Chen, Chao Yang
Proof: Let q(τ)= τ, g(τ) = FT(τ), τ[tmin, tmax], where tmin and tmax are the lower and upper bounds of travel time. q(τ) and g(τ) are continuous in the interval [tmin, tmax]. Note that the CDF of travel time is a continuous function. According to the property of continuous function, if two functions are continuous in some domain, their summation or multiplication is also continuous in this domain. Thus, E[Tτ] in Equation (1) is still a continuous function. The extreme value theorem states that if a real-valued function is continuous in a closed interval [a, b], then the function attains its maximum and minimum in this domain (Rudin 1976). In our context, there always exists a travel time τ1 in [tmin, tmax] such that: That is, CET always exists. This completes the proof.
Low level formation controls for a group of quadrotors with model uncertainties
Published in International Journal of Control, 2020
Select now , and let . Note that for we have . Fix with and consider the derivative . Clearly for all u. For we have (since we consider now ) and then . For we have and then . By the extreme value theorem the continuous function attains its absolute minimum in the interval . Since the minimum of the function on the same interval is the same, i.e. in . Hence, we have for all . Select now such that . Since we finally conclude that for all the derivative of V in (42) satisfies for all . This also confirms that for all . Note that from the above considerations can be taken as small as needed and so is with (by increasing gain a). In conclusion we have shown that in case (ii) the solution of (37) is globally uniformly bounded. Moreover, the solution of the system is globally uniformly ultimately bounded with the ultimate bound .