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David G. Luenberger
Professor
Management Science and Engineering
Office: Terman 410 | Phone: 650-723-3039 | Fax: 650-723-1614
Email: luen @ stanford.edu
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Research Interests and Highlights
Prof. Luenberger's long-term interest is
the role of mathematics in the solution of important problems
of planning, decision, operations, and strategy. A general theme
of his interest is "Better Living Through Mathematics."
He has worked on numerous practical problems, but his main objective
has been the development of theory. The principal areas of his
theoretical work and interests are highlighted below.
- Control Theory: (1961-80)
Prof. Luenberger developed the Luenberger Observer in his Ph.D.
dissertation. An observer constructs the state of a dynamic
system from observations of a few of the system's outputs. Observers
are used extensively in modern control design. An outgrowth
of this research was the Luenberger Canonical Form used to form
convenient represenations of complex dynamic systems. He also
carried out research on optimal control and dynamic games.
The theory of descriptor variable systems
was initated by Prof Luenberger. This theory extends common
state-space theory to systems described by difference equations
or differential equations that contain static relations. Some
of this work is reported in the text Introduction
to Dynamic Systems.
- General Optimization:
(1964-70) Theory:
Prof. Luenberger pioneered in the development of otpimization
in abstract linear spaces. Most of this research was reported
in the book Optimization by Vector
Space Methods. Of particular importance in this area is
the general theory of duality for optimization which in many
cases converts a problem in high dimention to one in low dimention.
The abstract theory is used in numerous fields, including, optimal
control, microeconomics, finance, eonometrics, accounting, and
numerical analysis.
- Mathematical Programming: (1970-84)
Research in this field focused on creation
of a theory that predicts the rate of convergence of mathematical
programming algorithms. This theory led to the concept of a
canonical rate that is an inherent property of an optimization
problem. Using this theory it is possible to predict the speed
at which an algorithm will converge and, more importantly, to
devise methods for changing the structure of the problem so
that convergence will be accelerated. The theory also led to
the development of new efficient algorithms. This theory is
largely reported in the text Linear and
Nonlinear Programming, 2nd Ed..
- Microeconomics: (1984-96)
Prof. Luenberger developed the benefit
approach to microeconomics, unifing much of microeconomics and
leading to important new results. In this theory a benefit function
is constructed for each individual, based on his or her utility
function. A major result is the zero-maximum principle, which
states that a Pareto efficient allocation corresponds to the
total benefit (over all individuals) being maximized and being
zero. Hence. Pareto efficiency corresponds to an optimization
problem.
A dual principle, the zero-minimum principle,
states that at an equilibrium the surplus (the dual of total
benefit) in the economy is minimized and is zero. This principle
forms the basis of an efficient means for actually computing
the equilibrium of an economy. This theory is reported in
the text Microeconomic Theory.
- Investment Science: (Current)
Prof. Luenberger's current research
is the application of systems methods to investment problems.
He has developed the textbook Investment
Science used in a fundamental graduate course. His research
focuses on evaluation of uncertain multiperiod cash flow streams,
design of optimal portfolios of investments (including projects),
and construction of hedging strategies.
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