Robert Vanderbei

• Ph.D. Applied Mathematics
• Professor (Full) at Princeton University

Many real-world problems could be modeled as linear programs except that some or all of the variables are constrained to be integers. Such problems are called integer programming problems.

In the previous chapter, we discussed what it means when the ratios computed to calculate the leaving variable are all nonpositive (the problem is unbounded). In this chapter, we take up the more delicate issue of what happens when some of the ratios are infinite (i.e., their denominators vanish).

So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between the primal and the dual problems.

In this chapter, we define an interior-point method for linear programming that is called a path-following method. Recall that for the simplex method we required a two-phase solution procedure.

In Chapter 18, we described and analyzed an interior-point method called the path-following algorithm. This algorithm is essentially what one implements in practice but as we saw in the section on convergence analysis, it is not easy (and perhaps not possible) to give a complete proof that the method converges to an optimal solution. If convergence...

In this chapter, we discuss implementation issues that arise in connection with the path-following method.

In this chapter, we shall study if not the most practical then certainly an elegant application of linear programming. The subject is called game theory, and we shall focus on the simplest type of game, called the finite two-person zero-sum game, or just matrix game for short.

In the previous chapter, we rewrote the simplex method using matrix notation. This is the first step toward our aim of describing the simplex method as one would implement it as a computer program. In this chapter, we shall continue in this direction by addressing some important implementation issues.

In the previous chapter, we saw that the simplex method (with appropriate pivoting rules to guarantee no cycling) will solve any linear programming problem for which an optimal solution exists. In this chapter, we investigate just how fast it will solve a problem of a given size.

In this chapter, we begin our study of an alternative to the simplex method for solving linear programming problems. The algorithm we are going to introduce is called a path-following method. It belongs to a class of methods called interior-point methods. The path-following method seems to be the simplest and most natural of all the methods in this...

In this chapter, we consider two related subjects. The first, called sensitivity analysis (or postoptimality analysis) addresses the following question: having found an optimal solution to a given linear programming problem, how much can we change the data and have the current partition into basic and nonbasic variables remain optimal? The second s...

This final chapter on network-type problems deals with finding the best design of a structure to support a specified load at a fixed set of points.

In this chapter, we discuss briefly the most important applications of network flow problems.

In this chapter, we shall study a few applications of linear programming to an area of statistics called regression and to an area of machine learning called support vector machines. As a specific example for our study of regression, we shall use size and iteration-count data collected from a standard suite of linear programming problems to derive...

In the previous chapter, we showed that the step direction for the path-following method can be decomposed into a linear combination of three directions: a direction toward optimality, a direction toward feasibility, and a direction toward centrality. It turns out that these directions, or minor variants of them, arise in all interior-point methods...

In Chapter 23, we studied a generalization of the linear programming problem in which variables were constrained to take on integer values. In this chapter, we consider a generalization of a different kind. Namely, we shall study the class of problems that would be linear programs except that the objective function is permitted to include terms inv...

Up until now, we have always considered our problems to be given in standard form. However, for real-world problems it is often convenient to formulate problems in the following form:

Many linear programming problems can be viewed as a problem of minimizing the “transportation” cost of moving materials through a network to meet demands for material at various locations given sources of material at other locations. Such problems are called network flow problems. They form the most important special class of linear programming pro...

Associated with every linear program is another called its dual. The dual of this dual linear program is the original linear program (which is then referred to as the primal linear program). Hence, linear programs come in primal/dual pairs. It turns out that every feasible solution for one of these two linear programs gives a bound on the optimal o...

The most time-consuming aspect of each iteration of the path-following method is solving the system of equations that defines the step direction vectors Δx, Δy, Δw, and Δz:

This book is mostly about linear programming. However, this subject, important as it is, is just a subset of a larger subject called convex analysis. In this chapter, we shall give a brief introduction to this broader subject. In particular, we shall prove a few of the fundamental results of convex analysis and see that their proofs depend on some...

In this chapter we present the simplex method as it applies to linear programming problems in standard form.

In this chapter, we shall study some applications of linear programming to problems in quantitative finance.

In the last chapter, we saw that small modifications to the primal–dual interior-point algorithm allow it to be applied to quadratic programming problems as long as the quadratic objective function is convex. In this chapter, we shall go further and allow the objective function to be a general (smooth) convex function. In addition, we shall allow t...

Future large space telescopes will be equipped with adaptive optics (AO) to overcome wavefront aberrations and achieve high contrast for imaging faint astronomical objects, such as Earth-like exoplanets and debris disks. In contrast to AO that is widely used in ground telescopes, space-based AO systems will use focal plane wavefront sensing to meas...

Future large space telescopes will be equipped with adaptive optics (AO) to overcome wavefront aberrations and achieve high contrast for imaging faint astronomical objects, such as Earth-like exoplanets and debris disks. In contrast to AO that is widely used in ground telescopes, space-based AO systems will use focal plane wavefront sensing to meas...

The book provides a broad introduction to both the theory and the application of optimization with a special emphasis on the elegance, importance, and usefulness of the parametric self-dual simplex method. The book assumes that a problem in “standard form,” is a problem with inequality constraints and nonnegative variables. The main new innovation...

A coronagraphic starlight suppression system situated on a future flagship space observatory offers a promising avenue to image Earth-like exoplanets and search for biomarkers in their atmospheric spectra. One NASA mission concept that could serve as the platform to realize this scientific breakthrough is the Large UV/Optical/IR Surveyor (LUVOIR)....

How ground-based telescopes schedule their observations in response to different variables can significantly impact their efficiency. Namely, competing science priorities and constraints, variations in the weather, and the visibility of a particular part of the sky are some of the determining variables. In this paper we introduce the Feature-based...

All coronagraphic instruments for exoplanet high-contrast imaging need wavefront correction systems to reject optical aberrations and create sufficiently dark holes. Since the most efficient wavefront correction algorithms (controllers and estimators) are usually model-based, the modeling accuracy of the system influences the ultimate wavefront cor...

How ground-based telescopes schedule their observations in response to competing science priorities and constraints, variations in the weather, and the visibility of a particular part of the sky can significantly impact their efficiency. In this paper we introduce the Feature-Based telescope scheduler that is an automated, proposal-free decision ma...

A coronagraphic starlight suppression system situated on a future flagship space observatory offers a promising avenue to image Earth-like exoplanets and search for biomarkers in their atmospheric spectra. One NASA mission concept that could serve as the platform to realize this scientific breakthrough is the Large UV/Optical/IR Surveyor (LUVOIR)....

All coronagraphic instruments for exoplanet high-contrast imaging need wavefront correction systems to reject optical aberrations and create sufficiently dark holes. Since the most efficient wavefront correction algorithms (controllers and estimators) are usually model-based, the modeling accuracy of the system influences the ultimate wavefront cor...

High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em regularization factor}, and solve them by the parametric simplex method (PSM). Our parametric simplex method o...

It is an interesting and counterintuitive fact that a Slinky released from a hanging position does not begin to fall all at once but rather each part of the Slinky only starts to fall when the collapsed part above it reaches its level. The analyses published so far have given physical arguments to explain this property of Slinkies. In particular, t...

For direct imaging of faint exoplanets, coronagraphs are widely used to suppress light and achieve a high contrast. Wavefront correction algorithms based on adaptive optics are introduced simultaneously to mitigate aberrations in the optical system. Stroke minimization is one of the primary control algorithms used for high-contrast wavefront contro...

An external flower-shaped occulter flying in formation with a space telescope can theoretically provide sufficient starlight suppression to enable direct imaging of an Earth-like planet. Occulter shapes are scaled to enable experimental validation of their performance at laboratory dimensions. Previous experimental results have shown promising perf...

Recent efforts combining the optimization techniques of apodized pupil Lyot coronagraphs (APLC) and shaped pupils have demonstrated the viability of a binary-transmission mask architecture for extremely high contrast (10-10) exoplanet imaging. We are now building on those innovations to carry out a survey of Lyot coronagraph performance for large,...

A starshade or external occu lter is a spacecraft flown along the line -of -sight of a space telescope to suppress starlight and enable high -contrast direct imaging of exoplanets. Because of its lar ge size and scale it is impossible to fully test a starshade system on the ground before launch. Therefore , laboratory verification of starshade desi...

Princeton University is upgrading our space occulter testbed. In particular, we are lengthening it to ~76m to achieve flightlike Fresnel numbers. This much longer testbed required an all-new enclosure design. In this design, we prioritized modularity and the use of commercial off-the-shelf (COTS) and semi-COTS components. Several of the technical c...

Feature-based Scheduler offers a sequencing strategy for ground-based telescopes. This scheduler is designed in the framework of Markovian Decision Process (MDP), and consists of a sub-linear online controller, and an offline supervisory control-optimizer. Online control law is computed at the moment of decision for the next visit, and the supervis...

We propose two approaches to solve large-scale compressed sensing problems. The first approach uses the parametric simplex method to recover very sparse signals by taking a small number of simplex pivots, while the second approach reformulates the problem using Kronecker products to achieve faster computation via a sparser problem formulation. In p...

Coronagraphs of the apodized pupil and shaped pupil varieties use the Fraunhofer diffraction properties of amplitude masks to create regions of high contrast in the vicinity of a target star. Here we present a hybrid coronagraph architecture in which a binary, hard-edged shaped pupil mask replaces the gray, smooth apodizer of the apodized pupil Lyo...

The baseline design for the WFIRST-AFTA Coronagraphic Instrument includes several shaped pupil masks. Our newest designs incorporate the shaped pupil into a Lyot-style architecture to access smaller inner working angles at a given tradeoff in throughput and contrast. We describe two mask configurations: a characterization mode for spectroscopy of e...

An external occulter is a spacecraft own along the line-of-sight of a space telescope to suppress starlight and enable high-contrast direct imaging of exoplanets. The shape of an external occulter must be specially designed to optimally suppress starlight; however, deviations from the ideal shape such as manufacturing errors can result in loss of s...

One of the main candidates for creating high-contrast for future Exo-Earth detection is an external occulter or sharshade. A starshade blocks the light from the parent star by flying in formation along the line-of-sight from a space telescope. Because of its large size and scale it is impossible to fully test a starshade system on the ground before...

In this paper, we derive explicit formulas for the distribution of the complex roots of random sums of the form $P_n(z) = \sum_{j=0}^{n} \eta_j f_j(z)$, where the $\eta_j$ are iid standard Gaussian random variables and the functions $f_j$ are given analytic functions that are real-valued on the real line. We consider some special cases including We...

In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (Evans-Steuer) algorithm [12]. Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions....

Given two vectors, not necessarily of the same length, each having increasing elements, we form the matrix whose (i, j)-th element is the sum of the i-th element from the first vector and the j-th element from the second vector. Such a matrix is called an outer sum of the two vectors (a concept that is analogous to outer products). If we assume tha...

The mineral assemblage of a cement paste may vary considerably with its environment. In addition, the water content of a cement paste is relatively low and the ionic strength of the interstitial solution is often high. These conditions are extreme conditions with respect to the common assumptions made in speciation problem. Furthermore the common t...

We describe the expected scientific capabilities of CHARIS, a high-contrast integral-field spectrograph (IFS) currently under construction for the Subaru telescope. CHARIS is part of a new generation of instruments, enabled by extreme adaptive optics (AO) systems (including SCExAO at Subaru), that promise greatly improved contrasts at small angular...

Several years ago at Princeton we invented a technique to optimize shaped pupil (SP) coronagraphs for any telescope aperture. In the last year, our colleagues at the Jet Propulsion Laboratory (JPL) invented a method to produce these non-freestanding mask designs on a substrate. These two advances allowed us to design SPs for two possible space tele...

An external occulter is a specially-shaped spacecraft own along the line-of-sight of a space telescope to block starlight before reaching its entrance pupil. Using optimization methods, occulter shapes can be designed to most effectively block starlight. A full-scale occulter cannot be tested on the ground and its performance must be predicted; the...

In this chapter we present the simplex method as it applies to linear programming problems in standard form.

This final chapter on network-type problems deals with finding the best design of a structure to support a specified load at a fixed set of points. The topology of the problem is described by a graph where each node represents a joint in the structure and each arc represents a potential member.1 We shall formulate this problem as a linear programmi...

In this chapter, we shall study if not the most practical then certainly an elegant application of linear programming. The subject is called game theory, and we shall focus on the simplest type of game, called the finite two-person zero-sum game, or just matrix game for short. Our primary goal shall be to prove the famous Minimax Theorem, which was...

In this chapter, we discuss briefly the most important applications of network flow problems.

In this chapter, we discuss implementation issues that arise in connection with the path-following method. The most important issue is the efficient solution of the systems of equations discussed in the previous chapter. As we saw, there are basically three choices, involving either the reduced KKT matrix, $$\displaystyle{ B = \left [\begin{array}{...

This book is mostly about a subject called Linear Programming. Before defining what we mean, in general, by a linear programming problem, let us describe a few practical real-world problems that serve to motivate and at least vaguely to define this subject.

We develop an R package fastclime for solving a family of regularized linear programming (LP) problems. Our package efficiently implements the parametric simplex algorithm, which provides a scalable and sophisticated tool for solving large-scale linear programs. As an illustrative example, one use of our LP solver is to implement an important spars...

Up until now, we have always considered our problems to be given in standard form. However, for real-world problems it is often convenient to formulate problems in the following form

In this chapter, we shall study some applications of linear programming to problems in quantitative finance.

An external occulter is a satellite employing a large screen, or starshade, that flies in formation with a spaceborne telescope to provide the starlight suppression needed for detecting and characterizing exoplanets. Among the advantages of using an occulter are the broadband allowed for characterization and the removal of light before entering the...

High contrast imaging requires focal plane wavefront control and estimation to correct aberrations in an optical system; non-common path errors prevent the use of conventional estimation with a separate wavefront sensor. The High Contrast Imaging Laboratory (HCIL) at Princeton has led the development of several techniques for focal plane wavefront...

This book is mostly about linear programming. However, this subject, important as it is, is just a subset of a larger subject called convex analysis. In this chapter, we shall give a brief introduction to this broader subject. In particular, we shall prove a few of the fundamental results of convex analysis and see that their proofs depend on some...

So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between the primal and the dual problems.

In the previous chapter, we discussed what it means when the ratios computed to calculate the leaving variable are all nonpositive (the problem is unbounded). In this chapter, we take up the more delicate issue of what happens when some of the ratios are infinite (i.e., their denominators vanish).

In this chapter, we shall study an application of linear programming to an area of statistics called regression. As a specific example, we shall use size and iteration-count data collected from a standard suite of linear programming problems to derive a regression estimate of the number of iterations needed to solve problems of a given size.

In Chapter 23, we studied a generalization of the linear programming problem in which variables were constrained to take on integer values. In this chapter, we consider a generalization of a different kind. Namely, we shall study the class of problems that would be linear programs except that the objective function is permitted to include terms inv...

One of the most promising concepts of starlight suppression for direct imaging of exoplanets is flying a specially-shaped external occulter in formation with a space telescope. Here we present contrast performance verification of an occulter design scaled to laboratory-size using Fresnel numbers corresponding to the space design. Experimental desig...

In this paper we present two new approaches to efficiently solve large-scale compressed sensing problems. These two ideas are independent of each other and can therefore be used either separately or together. We consider all possibilities. For the first approach, we note that the zero vector can be taken as the initial basic (infeasible) solution f...

The recently completed study of using one of the AFTA telescopes for a potential WFIRST mission included a coronagraph instrument for exoplanet imaging. The challenge is to design a coronagraph that achieves the desired high contrast in the presence of the complicated on-axis optical architecture of the AFTA. This is especially difficult if contras...

To spectrally characterize Earth-like planets around nearby stars with a coronagraph, an extreme adaptive optics (ExAO) system is mandatory. The correction of the amplitude and phase aberrations in the wavefront on both sides of the image plane and in sufficiently large bandwidths can be done with two deformable mirrors (DM) in a pupil mapping conf...

The construction presented in this paper can be briefly described as follows: starting from any "finite-dimensional" Markov transition function p_t, on a measurable state space (E,B), we construct a strong Markov process on a certain "intrinsic" state space that is, in fact, a closed subset of a finite dimensional Euclidean space R^d. Of course we...

Imaging planets orbiting nearby stars requires a system for suppressing the host starlight by at least ten orders of magnitude. One such approach uses an external occulter, a satellite flying far from the telescope and employing a large screen, or starshade, to suppress the incoming starlight. This trades the added complexity of building the precis...

An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual...

The National Reconnaissance Office (NRO) recently made available two 2.4-m telescope assemblies to NASA for astrophysical missions. These open up the possibility of achieving the scientific goals of the WFIRST mission proposed by the "New Worlds, New Horizons" decadal survey. These telescopes are large enough to support a scientifically powerful co...

Direct imaging of an Earth-like planet around nearby stars represents a challenge for two primary reasons. First, the intensity ratio between the bright star and its dim Earth-like planetary companion is expected to be approximately ten orders of magnitude and secondly the angular separation to the star is very small. An external occulter is a spec...

The aim of this chapter is to discuss several applications of metric space ideas to some classical problems of engineering analysis.

This chapter is about the problem of optimizing a convex function. The concept of duality is introduced and connections with Lagrangians are made. Throughout, a convex function is said to be proper if its epigraph is not empty, that is, the function does take some real values. Also, by an abuse of language, it will be said to be closed if its epigr...

This introductory chapter is devoted to general notions regarding sets, functions, sequences, and series. We aim to introduce and review the basic notation, terminology, conventions, and elementary facts.

Linear functions are the simplest functions in linear spaces. Their study is of fundamental importance and forms the subject known as linear algebra. In this chapter we assume the reader has mastered the essentials of linear algebra. We therefore spend some time on the class of functions and sets that are just one step more complicated than the lin...

Basic questions of analysis on the real line are tied to the notions of closeness and distances between points. The same issue of closeness comes up in more complicated settings, for instance, when we try to approximate a function by a simpler function. Our aim is to introduce the idea of distance in general so that we can talk of the distance betw...

For mappings from one metric space to another we employ either notations like T, S, U or notations like f, g, h. Generally, the transformation notation is cleaner: we write Tx for the image of x under T, which becomes f(x) in the standard function notation.

In this paper, we consider an issue of great interest to all students: fairness in grading. Specifically, we represent each grade as a student's intrinsic (overall) aptitude minus a correction representing the course's inherent difficulty plus a statistical error. We consider two statistical methods for assigning an aptitude to each student and, si...

Using 55 years of daily average temperatures from a local weather station, I made a least-absolute-deviations (LAD) regression model that ac-counts for three effects: seasonal variations, the 11-year solar cycle, and a linear trend. The model was formulated as a linear programming problem and solved using widely available optimization software. The...

Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. It is well-known that the fast Fourier transform (fft) is a recursive algorithm that can dramatically improve the efficiency for computing the discrete Fou...