introduction to the theory of nonlinear optimization

[...]... complete this introduction with a short compendium of the structure of this textbook Of course, the question of the solvability of a concrete nonlinear optimization problem is of primary interest and, therefore, existence theorems are presented in Chapter 2 Subsequently the question about characterizations of minimal points runs like a red thread through this book For the formulation of such characterizations... to the original problem An apphcation of optimality conditions and duahty theory to semidefinite optimization being a topical field of research in optimization, is described in Chapter 7 The results in the last chapter show that solutions or characterizations of solutions of special optimization problems with a rich mathematical structure can be derived sometimes in a direct way It is interesting to. .. weakly to some x E S Because of the weak lower semicontinuity of / it follows f{x) < liminf/(xnj = inf/(:^), and the theorem is proved D Now we proceed to specialize the statement of Theorem 2.3 in order to get a version which is useful for apphcations Using the concept of the epigraph we characterize weakly lower semicontinuous functionals Definition 2.4 Let the assumption (2.1) be satisfied The set... couvcrging to some u E S, Then we obtain ti f{un)-f{u) = J[g{xo + L{un){t))+g{xo + L{u){t))]dt to ti + f[h{Un{t))-h{u{t))]dt (2.11) to Because of the inequality (2.10) and the continuity of g the following equation holds pointwise: lim g{xo + L{un){t)) = g{x^ + L{u){t)) n—>oo Since ||t^n||L5^ [to, tii < 1 and ||t^||Lj^ [to, tii < 1, the convergence of the first integral in (2.11) to 0 follows from Lebesgue's theorem... answering the question about the existence of a minimal solution of an optimization problem, in this section the set of all minimal points is investigated Theorem 2.14 Let S be a nonempty convex subset of a real linear space For every quasiconvex functional f : S -^ R the set of minimal points of f on S is convex Proof If / has no minimal point on S, then the assertion is evident Therefore we assume that... and let the system of differential equations be given as x{t) = Ax{t) + Bu{t) almost everywhere on [to, ti] (2.8) with the initial condition x (to) = xo E M^ (2.9) where — oo < to < ^i < oo Let the control i/ be a 1/2^ [to, ^i] function A solution X of the system (2.8) of differential equations with the initial condition (2.9) is defined as t x{t) =xo+ f e^^^-'^Bu{s) ds for all t G [to, h] to The exponential... known as Euler-Lagrange equation for a minimal solution of the problem in Example 1.2 The Pontryagin maximum principle is the essential tool for the solution of the optimal control problem formulated in Example 1.3; an optimal control is determined in the Examples 5.21 and 5.23 An application of the alternation theorem leads to a solution of the linear Chebyshev approximation problem (given in Example... First notice that X := L^ [to, ii] is a reflexive Banach space Since S is the closed unit ball in L2^ [to, ti], the set S is closed, bounded and convex Next we show the quasiconvexity of the objective functional / For that purpose we define the linear mapping L : 5 — A(7^ [to, ii] (let AC^ [to^ ti] denote the real linear space of ab> solutely continuous n vector functions equipped with the maximum norm) with... For the proof of the continuity of / at £ we take any s G (0,1) Then we choose an arbitrary element x of the closed ball B{x,€{l — j)g)' Because of the convexity of / we get for some y G 5(Ox, (1 — j)g) f{x) = f{x + ey) 2.2 Existence Theorems 17 = f{{l-s)x + s{x + y)) < {l-e)f{x)+sf{x + y) < {l-e)f{x)+€p which imphes f{x)-f{x) . alt="" Introduction to the Theory of Nonhnear Optimization Johannes Jahn Introduction to the Theory of NonHnear Optimization Third Edition With 31 Figures Sprin g er Prof. Dr complete this introduction with a short compendium of the structure of this textbook. Of course, the question of the solvability of a concrete nonlinear optimization problem is of primary interest. interesting to note that the Hahn-Banach theorem (often in the version of a separation theorem like the Eidelheit separation theo- rem) proves itself to be the key for central characterization theorems.