By David Yang Gao
Encouraged by way of useful difficulties in engineering and physics, drawing on quite a lot of utilized mathematical disciplines, this e-book is the 1st to supply, inside of a unified framework, a self-contained finished mathematical idea of duality for common non-convex, non-smooth platforms, with emphasis on tools and purposes in engineering mechanics. issues lined contain the classical (minimax) mono-duality of convex static equilibria, the attractive bi-duality in dynamical platforms, the attention-grabbing tri-duality in non-convex difficulties and the advanced multi-duality regularly canonical platforms. A almost certainly strong sequential canonical twin transformation process for fixing totally nonlinear difficulties is built heuristically and illustrated through use of many attention-grabbing examples in addition to vast purposes in a large choice of nonlinear structures, together with differential equations, variational difficulties and inequalities, restricted international optimization, multi-well section transitions, non-smooth post-bifurcation, huge deformation mechanics, structural restrict research, differential geometry and non-convex dynamical platforms. With exceedingly coherent and lucid exposition, the paintings fills a huge hole among the mathematical and engineering sciences. It exhibits tips to use formal language and duality easy methods to version typical phenomena, to build intrinsic frameworks in several fields and to supply rules, strategies and strong equipment for fixing non-convex, non-smooth difficulties bobbing up obviously in engineering and technological know-how. a lot of the ebook includes fabric that's new, either in its demeanour of presentation and in its learn improvement. A self-contained appendix offers a few priceless heritage from ordinary sensible research. viewers: The publication might be a useful source for college students and researchers in utilized arithmetic, physics, mechanics and engineering. the entire quantity or chosen chapters is usually advised as a textual content for either senior undergraduate and graduate classes in utilized arithmetic, mechanics, basic engineering technological know-how and different parts within which the notions of optimization and variational equipment are hired.
Read Online or Download Duality Principles in Nonconvex Systems - Theory, Methods and Applications (Nonconvex Optimization and its Applications, Volume 39) PDF
Best linear programming books
"Optimization thought is turning into a a growing number of very important mathematical in addition to interdisciplinary sector, specifically within the interaction among arithmetic and lots of different sciences like machine technology, physics, engineering, operations examine, and so on. "This quantity offers a complete advent into the idea of (deterministic) optimization on a complicated undergraduate and graduate point.
This is often the 1st complete reference on trust-region equipment, a category of numerical algorithms for the answer of nonlinear convex optimization equipment. Its unified therapy covers either unconstrained and limited difficulties and stories a wide a part of the really expert literature at the topic.
On hand for the 1st time in paperback, R. Tyrrell Rockafellar's vintage learn offers readers with a coherent department of nonlinear mathematical research that's particularly suited for the research of optimization difficulties. Rockafellar's idea differs from classical research in that differentiability assumptions are changed via convexity assumptions.
Hybrid dynamical platforms convey non-stop and immediate adjustments, having positive aspects of continuous-time and discrete-time dynamical platforms. packed with a wealth of examples to demonstrate strategies, this publication offers a whole concept of sturdy asymptotic balance for hybrid dynamical platforms that's acceptable to the layout of hybrid regulate algorithms--algorithms that function good judgment, timers, or combos of electronic and analog elements.
- The SIAM 100-digit challenge. Study in high-accuracy numerical computing
- Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems
- Topological Methods in Complementarity Theory
- The Stability of Matter: From Atoms to Stars: Selecta of Elliot H. Lieb
Additional resources for Duality Principles in Nonconvex Systems - Theory, Methods and Applications (Nonconvex Optimization and its Applications, Volume 39)
On the other hand, the minimals x(t) and y(t) have to intersect transversely at a point t = τ . Otherwise they would coincide according to the uniqueness theorem of differential equations. The means however, that there exists < 0, so that the path t → x(t) is not minimal on the interval [τ − , T ] for large enough T . Therefore • On the interval [τ − , τ + ] the action can be decreased by a fixed positive value if a minimal C 2 -path q(t) is chosen from x(t − ) to y(t + ) instead taking on [τ − , τ + ] x(t) and y(t) and go around corners.
Linearize the Euler equations for F = ab + cos(2πv))2 + (v )2 ). 40 CHAPTER 1. ONE-DIMENSIONAL VARIATIONAL PROBLEMS 12) Show that the geodesic in example 4) which is given by I = r sin(ψ) defines an extremal field if −(a − b) < c < a − b. Discuss the geodesic for c = a − b, for a − b < c < a + b and for c = a + b. 1 Global extremal fields The two-dimensional torus has the standard representation T 2 = R2 /Z2 . We often will work on his covering surface R2 , where everything is invariant under his fundamental group Z2 .
4. Therefore i) to iii) are equivalent with i) ii) iii) ∃M, |ξn (t)| ≤ M, t ∈ [0, T ] |ξn+1 (t) − ξn (t)| → 0, n → ∞, t ∈ [0, q] ˙ |ξ˙n+1 (t) − ξ(t)| → 0, n → ∞, t ∈ [0, q] . The claim i)’ has already been proven by giving the periodic function ξ(t). 4 we see that |ξn (t)| ≤ |ξ˙n (t)| ≤ C1 C2 and this means, that ξn (t) and ξ˙n (t) are quicontinuous uniformly bounded sequences of functions. According to the theorem of Arzela-Ascoli, they converge uniformly. So, also (ii) and (iii) are proven. ✷ As a corollary of this theorems we see, that if γ ∈ M and γ is not periodic, one has an order on Z2 defined by (j, k) < (j , k ) if x(t + j) − k < x(t + j ) − k , ∀t .