Alexander Shapiro

Optimization problems involving sequential decisions in a stochastic environment were studied in Stochastic Programming (SP), Stochastic Optimal Control (SOC) and Markov Decision Processes (MDP). This monograph concentrates on SP and SOC modeling approaches. In these frameworks, there are natural situations when the considered problems are convex. The classical approach to sequential optimization is based on dynamic programming. It has the problem of the so-called “curse of dimensionality”, in that its computational complexity increases exponentially with respect to the dimension of state variables.Recent progress in solving convex multistage stochastic problems is based on cutting plane approximations of the cost-to-go (value) functions of dynamic programming equations. Cutting plane type algorithms in dynamical settings is one of the main topics of this monograph. Also discussed in this work are stochastic approximation type methods applied to multistage stochastic optimization problems. From the computational complexity point of view, these two types of methods seem to be complimentary to each other. Cutting plane type methods can handle multistage problems with a large number of stages but a relatively smaller number of state (decision) variables. On the other hand, stochastic approximation type methods can only deal with a small number of stages but a large number of decision variables.

Covers optimization problems involving uncertain parameters for which stochastic models are available. These problems occur in almost all areas of science and engineering.

In this book on Richard Wagner’s compelling but enigmatic masterpiece Götterdämmerung, the final opera of his monumental Ring tetralogy, Alexander H. Shapiro advances an ambitious new interpretation which uncovers intriguing new facets to the work’s profound insights into the human condition. By taking a fresh look at the philosophical and historical influences on Wagner, and critically reevaluating the composer’s intellectual worldview as revealed in his own prose works, letters, and diary entries, the book challenges a number of conventional views that continue to impede a clear understanding of this work’s meaning. The book argues that Götterdämmerung, and hence the Ring as a whole, achieves coherence when interpreted in terms of contemporary nineteenth-century theories of progress, and, in particular, G.W.F. Hegel’s philosophies of mind and history. A central target of the book is the article of faith that has come to dominate Wagner scholarship over the years – that Wagner’s encounter in 1854 with Arthur Schopenhauer’s philosophy conclusively altered the final message of the Ring from one of historical optimism to existential pessimism. The author contends that Schopenhauer’s uncompromising denigration of the will and denial of the possibility for human progress find no place in the written text of the Ring or in a plausible reading of the final musical setting. In its place, the author discovers in the famous Immolation Scene a celebration of mankind’s inexhaustible capacity for self-improvement and progress. The author makes the further compelling case that this message of progress is communicated not through Siegfried, the traditional male hero of the drama, but through Brünnhilde, the warrior goddess who becomes a mortal woman. In her role as a battle-tested world-historical prophet she is the true revolutionary change agent of Wagner’s opera who has the strength and vision to comprehend and thereby shape human history. This highly lucid and accessible study is aimed not only at scholars and researchers in the fields of opera studies, music and philosophy, and music history, but also Wagner enthusiasts, and readers and students interested in the history and philosophy of the nineteenth century.

In this book on Richard Wagner’s compelling but enigmatic masterpiece Götterdämmerung, the final opera of his monumental Ring tetralogy, Alexander H. Shapiro advances an ambitious new interpretation which uncovers intriguing new facets to the work’s profound insights into the human condition. By taking a fresh look at the philosophical and historical influences on Wagner, and critically reevaluating the composer’s intellectual worldview as revealed in his own prose works, letters, and diary entries, the book challenges a number of conventional views that continue to impede a clear understanding of this work’s meaning. The book argues that Götterdämmerung, and hence the Ring as a whole, achieves coherence when interpreted in terms of contemporary nineteenth-century theories of progress, and, in particular, G.W.F. Hegel’s philosophies of mind and history. A central target of the book is the article of faith that has come to dominate Wagner scholarship over the years – that Wagner’s encounter in 1854 with Arthur Schopenhauer’s philosophy conclusively altered the final message of the Ring from one of historical optimism to existential pessimism. The author contends that Schopenhauer’s uncompromising denigration of the will and denial of the possibility for human progress find no place in the written text of the Ring or in a plausible reading of the final musical setting. In its place, the author discovers in the famous Immolation Scene a celebration of mankind’s inexhaustible capacity for self-improvement and progress. The author makes the further compelling case that this message of progress is communicated not through Siegfried, the traditional male hero of the drama, but through Brünnhilde, the warrior goddess who becomes a mortal woman. In her role as a battle-tested world-historical prophet she is the true revolutionary change agent of Wagner’s opera who has the strength and vision to comprehend and thereby shape human history. This highly lucid and accessible study is aimed not only at scholars and researchers in the fields of opera studies, music and philosophy, and music history, but also Wagner enthusiasts, and readers and students interested in the history and philosophy of the nineteenth century.

The main subject of this book is perturbation analysis of continuous optimization problems. In the last two decades considerable progress has been made in that area, and it seems that it is time now to present a synthetic view of many important results that apply to various classes of problems. The model problem that is considered throughout the book is of the form (P) Min/(x) subjectto G(x) E K. xeX Here X and Y are Banach spaces, K is a closed convex subset of Y, and / : X -+ IR and G : X -+ Y are called the objective function and the constraint mapping, respectively. We also consider a parameteriZed version (P ) of the above u problem, where the objective function / (x, u) and the constraint mapping G(x, u) are parameterized by a vector u varying in a Banach space U. Our aim is to study continuity and differentiability properties of the optimal value v(u) and the set S(u) of optimal solutions of (P ) viewed as functions of the parameter vector u.

The main subject of this book is perturbation analysis of continuous optimization problems. In the last two decades considerable progress has been made in that area, and it seems that it is time now to present a synthetic view of many important results that apply to various classes of problems. The model problem that is considered throughout the book is of the form (P) Min/(x) subjectto G(x) E K. xeX Here X and Y are Banach spaces, K is a closed convex subset of Y, and / : X -+ IR and G : X -+ Y are called the objective function and the constraint mapping, respectively. We also consider a parameteriZed version (P ) of the above u problem, where the objective function / (x, u) and the constraint mapping G(x, u) are parameterized by a vector u varying in a Banach space U. Our aim is to study continuity and differentiability properties of the optimal value v(u) and the set S(u) of optimal solutions of (P ) viewed as functions of the parameter vector u.