Davidson K.R., Donsig A.P. Real Analysis with Real Applications

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16.10 The Minimax Theorem 613

f may be written as a sum of squares. Thus f tends to inﬁnity as kxk goes to

inﬁnity. Therefore, the constraint set could be replaced with a compact set. Then

the Extreme Value Theorem asserts that the minimum is attained.

The constraint condition is really m linear conditions hx, A

i − a

≤ 0 for

1 ≤ j ≤ m, where a = (a

, . . . , a

) with respect to the standard basis e

, . . . , e

If rankA = m, then m ≤ n and A maps R

onto R

. Thus there are strictly

feasible points and so Slater’s condition is satisﬁed. In general this needs to be

checked.

The Lagrangian is deﬁned on R

× R

L(x, y) = f(x) +

j=1

hx, A

i − a

= hx, Qxi + hx, q + A

yi − ha, yi.

To ﬁnd a solution to the dual problem, we ﬁrst must compute h(y) = inf

x∈R

L(x, y).

This was solved in Example 16.7.12, so

h(y) = −



q + A

y, Q

−1

(q + A

− ha, yi

= −

hy, AQ

−1

yi − hy, a +

−1

qi −

hq, Q

−1

qi.

The dual problem is to maximize h(y) over the set R

. This is now a quadratic

programming problem with a simpler set of constraints, possibly at the expense of

extra variables if m > n. The matrix AQ

−1

is positive semideﬁnite but may not

be invertible. This is not a serious problem.

Now let’s look at a speciﬁc case.

Minimize f(x

, x

) = 2x

− 2x

+ 2x

− 6x

subject to x

≥ 0, x

≥ 0 and x

+ x

≤ 2.

This is a quadratic programming problem with Q =

2 −1

−1 2

, q = (−6, 0),

A =





−1 0

0 −1

1 1





, and a = (0, 0, 2). Note that Slater’s condition is satisﬁed, for

example, at the point (1/2, 1/2).

We can compute Q

−1

2 1

1 2

and

−1





2 1 −3

1 2 −3

−3 −3 6





, and

also a +

−1

q = (2, 1, −1) and

hq, Q

−1

qi = 6. Thus h(y

, y

) =

(2y

+ 2y

+ 6y

+ 2y

− 6y

) − 2y

− y

+ y

− 6.

This problem can be solved most easily using the (KKT) conditions. It will

be left to Exercise 16.10.G to show that they are satisﬁed for x = (3/2, 1/2) and

y = (0, 0, 1). Notice that f(3/2, 1/2) = −11/2 = h(0, 0, 1). Since the minimum

of f and the maximum of h are equal, this is the minimax value. Hence the value

614 Convexity and Optimization

of the program must be −11/2, the minimizer is (3/2, 1/2) and the multiplier is

(0, 0, 1).

Exercises for Section 16.10

A. Compute p

∗

and p

∗

for p(x, y) = sin(x + y) on R

B. Show that the set Y deﬁned in the proof of Lemma 16.10.2 is convex and open.

C. Show that the set A constructed in the proof of the Minimax Theorem (compact case)

is compact and convex.

D. Modify the proof of the Minimax Theorem to deal with the case in which X is un-

bounded and Y is compact.

E. Let p(x, y) = e

−x

− e

−y

(a) Show that p is convex–concave.

(b) Show that p

∗

= p

∗

(d) Why does this not contradict the Minimax Theorem?

F. Suppose that p(x, y) is convex–concave on X × X for a compact subset X of R

and

satisﬁes p(y, x) = −p(x, y). Prove that p

∗

= p

∗

= 0.

G. (a) Solve the (KKT) equations for the numerical example in Example 16.10.6.

(b) Write down the Lagrangian and verify the saddlepoint.

H. Consider the linear programming problem: Minimize hx, qi subject to Ax ≤ a,

where q ∈ R

, A is an m × n matrix, and a ∈ R

(a) Express this problem as a convex program and compute the Lagrangian.

(b) Find the dual program.

then the dual program has a solution w and hv, qi = hw, ai.

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Calif., 1989.

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Articles

[42] J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, “On Devaney’s Deﬁnition of Chaos,”

Amer. Math. Monthly 99 (1992), 332–334.

[43] H. Carslaw, “A Historical Note on Gibbs’ Phenomenon in Fourier’s Series and Integrals,” Bull.

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Amer. Math. Monthly 98 (1991), 47–49.

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[47] M. Vellekoop and R. Berglund, “On Intervals, Transitivity = Chaos,” Amer. Math. Monthly 101

(1994), 353–355.

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∞

−x

dx,

∞

cos

) dx and

∞

sin

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Amer. Math. Monthly 97 (1990), 39–42.

INDEX

A boldface page number indicates a deﬁnition or primary entry.

Abel, 32

Abel’s Test, 87

absolutely convergenet series, 81

absolutely integrable functions, 430

Accessibility Lemma, 566

afﬁne function, 142, 558

afﬁne hull, 559

afﬁne set, 558

afﬁnely dependent, 563

afﬁnely independent, 563

aleph nought (ℵ

), 61

algebra, 326

algebraic number, 65

algebraically closed ﬁeld, 455

almost everywhere (a.e.), 176

alternating sequence, 72

Alternating Series Test, 72

analytic function, 455

antiderivative, 165

antisymmetric, 26

Archimedes, 32

arithmetic mean–geometric mean inequality,

25, 582

generalized, 578

arithmetic–geometric mean, 95

Arzela–Ascoli Theorem, 243

Ascoli, 243

asymptotic cone, 563

asymptotic to a curve, 117, 147

attractive ﬁxed point, 332

attractive periodic point, 349

Axiom of Choice, 64, 202

B-spline wavelets, 554

Bailey, 32

Baire Category Theorem, 254, 362

ball, 248

Banach Contraction Principle, 338

Banach space, 185

Banach’s Isomorphism Theorem, 549

Banach–Steinhaus Theorem, 257

basis, 14

Battle–Lemari

e wavelets, 554

Bernoulli, D., 449

Bernstein, 63, 503

Bernstein polynomials, 290

Bernstein’s inequality, 504

Bernstein’s Theorem, 505

Bessel function, 225

Bessel’s DE, 415, 429

Bessel’s inequality, 198

bifurcation, 358

big O notation, 281

big tent map, 369

bijection, 8

bilateral shift, 543

binomial coefﬁcient, 24

fractional, 237

binomial series, 236

Binomial Theorem, 25

Bipolar Theorem, 597

Birkhoff Transitivity Theorem, 362

Bolzano–Weierstrass Theorem, 52

Borel, 103

Borel–Lebesgue Theorem, 175–177,185, 251,

328, 411

Borwein, P., 32

boundary, 564

bounded, 42, 102, 549

bounded above, 42

bounded below, 42

C, 454

dynamical system, 333

Cantor, 63, 64

Cantor function, 137

Cantor set, 103, 354, 367, 370, 385

generalized, 364

Cantor’s Intersection Theorem, 103, 254

Cantor’s Theorem, 65

Carath

eodory’s Theorem, 561, 573

cardinality, 60

Carleson, 471

Cartesian product, 7

Cauchy, 32, 38, 69

Cauchy Criterion for series, 69

Cauchy sequence, 56, 93, 184, 248

Cauchy–Schwarz inequality, 188

ceiling function, 116

Ces

aro means, 488

617

618 Index

chain rule, 145

change of variables, 167

chaos, 366

period doubling route to, 359

chaotic dynamical system, 366

characteristic function, 118

Chebychev polynomial, 307, 468, 498

Chebychev series, 311, 501

closed, 96, 185, 248

closed convex hull, 559

closed span, 200

closed under addition and scalar multiplica-

tion, 12

closure, 97

cluster point, 101, 104, 354. See also limit

point

cluster set, 354

codimension one, 298

codomain, 7

Cohen, 64

common reﬁnement, 154

compact, 101, 186, 251

renamed sequentially compact, 250

Comparison Test, 71

complement, 6

complete metric space, 248

complete normed vector space, 185

complete subset, 56, 93

completeness

of `

, 199

of C([a, b], R

), 398

of C([a, b], V ), 399

of C(K), 219

of C

(X), 249

of K(X), 380

of R, 56

of R

, 94

completion, 261

complex Fourier series, 462

complex numbers, 454

composition, 8, 122

concave function, 151, 576

conditionally convergenet series, 81

cone, 558

conjugate, 454

connected, 258

content zero, 175

Continuation Theorem, 408

continued fraction, 58

continuous, 248

continuous at a point, 109, 121

continuous at inﬁnity, 429

continuous dependence on parameters, 417

continuous extension, 262

continuous on a set, 109, 121

continuum hypothesis, 64

contraction, 336

contrapositive, 3

converge, 248

convergence tests for series

Abel’s Test, 87

Alternating Series Test, 72

Comparison Test, 71

Dirichlet’s Test, 85

Hardy’s Tauberian Theorem, 494

Integral Test, 76

Limit Comparison Test, 75

Ratio Test, 75

Root Test, 72

converges, 38, 66, 92, 184

converges pointwise, 213

converges uniformly, 214

convex, 152

convex function, 151, 563, 576

convex hull, 559

convex program, 602

Convex Projection Theorem, 569

convex set, 557

convex–concave function, 608

coordinate functions, 112

cosine law, 92

countable set, 61

counterexample, 3

cubic spline, 315, 555

d’Alembert, 37, 423, 449

Daniell integral, 267

Darboux’s Theorem, 152

Daubechies wavelets, 528

decreasing function, 135

Dedekind cuts, 33

degree of a trigonometric polynomial, 495

dense, 45, 101

derivative

vector-valued function, 391

diagonalization, 63, 243

diagonally dominant, 318

differentiable

vector-valued function, 391

differentiable at a point, 141

differentiable on an interval, 141

differential, 587

differential equation of nth order, 395

dimension, 15, 559

Index 619

Dini’s Test, 484

Dini’s Theorem, 217, 232

Dini–Lipschitz Theorem, 512

direct sum (⊕), 18, 521

directional derivative, 589

Dirichlet kernel, 476

Dirichlet’s Test, 85

Dirichlet–Jordan Theorem, 482, 496

discontinuous, 109

discrete dynamical system, 331

discrete metric, 247

diverge, 66

Divergence Theorem, 424

domain, 7

Dominated Convergence Theorem, 271

dot product, 89

doubling map, 353

du Bois Reymond, 471

dual program, 612

Dufﬁn’s duality gap, 608

dyadic interval, 516

dyadic wavelet, 514

empty interior, 99

epigraph, 563, 578

equicontinuous at a point, 240

equicontinuous on a set, 240

equioscillation condition, 301

equivalence class, 26

equivalence relation, 25

equivalent metrics, 249

equivalent statements, 2

error function, 294

error of approximation, 495

Euclidean norm, 89

Euler, 442, 449, 474

Euler’s constant, 50

even function, 146

eventually periodic decimal, 20

eventually periodic point, 350

exponential function (on C), 455

exposed face, 575

Extension Theorem, 262

extreme point, 573

Extreme Value Theorem, 125

face, 573

exposed, 575

Farkas Lemma, 573

father wavelet, 521

feasible point, 601

feasible vector, 602

Fej

er, 471, 488

Fej

er kernel, 489

Fej

er’s Theorem, 328, 490

Fermat’s Principle, 606

Fermat’s Theorem, 148

Fibonacci sequence, 23, 25, 238

ﬁeld, 31, 455

ﬁlter, 528

ﬁnite dimensional, 14

ﬁnite intersection property, 251

ﬁnite subcover, 251

ﬁrst category, 254

ﬁrst moment, 528

forcing term, 412

forward orbit, 331

Fourier, 423

Fourier coefﬁcients, 194

Fourier series, 194, 428

complex, 462

for L

functions, 466

fractal dimension, 385

fractals, 378

fractional binomial coefﬁcient, 237

Fraenkel, 64

Franklin wavelet, 541

function, 7, 481

convex, 576

Fundamental Theorem of Algebra, 455

Fundamental Theorem of Calculus, 164, 393

fundamental vibration, 448

set, 257

odel, 64

Galois, 32

gamma function, 174

Gauss, 38

Gaussian elimination, 18

generalized Cantor set, 364

geodesic, 246

geometric convergence, 345

geometric sequence, 71

geometric series, sum of, 71

Gibbs’s phenomenon, 485, 514

global minimizer, 584

Global Picard Theorem, 401

gradient, 586, 587

Gram–Schmidt Process, 192, 195

graph of the function, 7

greatest lower bound, 46

Green’s Theorem, 469

Haar coefﬁcients, 516

Haar system, 515

Haar wavelet, 518

620 Index

Hadamard’s Theorem, 233

half-spaces, 562

Hardy’s Tauberian Theorem, 494

harmonic extension, 434

harmonic function, 434

harmonic series, 66

harmonics, 446, 448

Harnack’s inequality, 439

hat function, 542

Hausdorff metric, 247

Heaviside function, 114

Heine–Borel Theorem, 103, 239

Helly’s Theorem, 564

Hermite, 32

Hessian matrix, 586

Hilbert space, 199

Hilbert–Schmidt norm, 191

older’s inequality, 209, 268

homeomorphic, 370

homeomorphism, 370

homogeneous linear DE, 412

horizontal line test, 8

Hurwitz, 469

hyperplane, 562

supporting, 572

ideal, 330

identity map (id

,id), 9, 17, 261

image, 7

imaginary part, 455

improper integral, 164, 223

inclusion map, 261

increasing function, 135

inﬁmal convolution, 586

inﬁmum, 46

inﬁnite decimal expansion, 34

inﬁnite series, 66

inﬂection points, 151

initial conditions, 395

initial value condition, 386

initial value problem, 395

injective, 8

inner product, 89, 187

integers, 6

integral

Daniell, 267

Lebesgue, 267

Riemann, 156

Integral Convergence Theorem, 220

integral equation, 387

Integral Test, 76

integrating factor, 403

integration by parts, 167

interior, 99, 564

Intermediate Value Theorem, 133

intersection, 6

inverse function, 8

inverse image, 8

isolated point, 104

isometric, 92

isometry, 261

isomorphic, 27

isoperimetric problem, 468

iterated function system, 378

itinerary, 374

Jackson, 503

Jackson’s Theorem, 506

Jensen’s inequality, 577

Jordan, 482

jump discontinuity, 115

Karush–Kuhn–Tucker Theorem, 603

Karush-Kuhn-Tucker conditions, 603

kernel, 16

Kuhn, 603

Lagrange, 423, 449

Lagrangian, 604

Lambert, 32

Laplace, 423

Laplacian, 424

Least Squares Theorem, 465

least upper bound, 46

Least Upper Bound Principle, 46

Lebesgue, 251, 471, 474

Lebesgue integral, 267

Lebesgue measure, 271

Lebesgue’s Theorem, 176, 225, 294

left-differentiable, 146

Legendre, 423

Leibniz, 32, 37, 72

Leibniz’s Rule, 222, 223

length of a regular curve, 394

limit, 38

Limit Comparison Test, 75

limit inferior, 50

limit of a function, 108

limit point, 96

limit superior, 50

Lindemann, 32

linear combination, 13

linear differential equation, 411

linear function, ambiguious meaning of, 142

linear programming, 614

Index 621

linear splines, 314

linear transformation, 15

linearly dependent, 14

linearly independent, 14, 563

Lipschitz at a point, 255

Lipschitz condition of order α (Lip α), 133,

498, 503

Lipschitz constant, 111

Lipschitz function, 111, 336

Lipschitz in the y-variable, 400

local minimizer, 584

Local Picard Theorem, 405

local solution, 405

local subgradient, 596

locally Lipschitz in the y-variable, 408

logistic functions, 355, 363

lower sum, 154

Lusin, 471

maple leaf, 385

matrix multiplication, 16

matrix representation, 15

maximal continuation, 409

maximin, 609

Maximum Principle, 443

mean value property, 445

Mean Value Theorem, 149, 391

Convex, 582

for integrals, 168

second-order, 322, 326

measurable function, 269

measure theory, 175, 266

measure zero, 105, 175

mesh, 154

method of undetermined coefﬁcients, 414

method, versus trick, 149

metric, 246

metric space, 246

minimal dynamical system, 354

minimax, 609

Minimax Theorem, 611

compact case, 610

minimizers, 584

Minkowski’s inequality, 211, 268

Minkowski’s Theorem, 575

Modiﬁed Newton’s method, 349

modulo 2π, 351

modulus, 454

modulus of continuity, 176, 294, 517

modus ponens, 2

Monotone Convergence Theorem, 47

Monotone Convergence Theorem (MCT), 268

monotone decreasing function, 135

monotone function, 135

monotone increasing function, 135

monotone increasing sequence, 47

Moreau–Yosida, 596

mother wavelet, 514

M-test, 227

multiplication operator, 550

multiplication operators, 545

multipliers, 603

multiresolution, 521

multiresolution analysis, 513

Riesz, 553

n-tuples, 7

negation, 2

Nested Intervals Lemma, 51

Newton, 32, 37, 344

Newton’s Method, 345

nodes, 314

nonhomogeneous linear DE, 412

nontrivial supporting hyperplane, 572

nonvertical hyperplane, 587

norm, 89, 179

normal cone, 596

normal derivative, 424

normed vector space, 179

not connected, 258

nowhere dense, 104, 254

nowhere differentiable functions, 230, 255

nowhere monotonic function, 257

nullity, 16

odd function, 117, 146

one-sided derivative, 146

one-sided limits, 115

one-to-one, 8

onto, 8

open, 98, 185, 248

open ball, 185

open cover, 251

open in S, 120

optimal solution, 602

optimal value, 602

order complete, 264

order dense, 264

order of a differential equation, 386

ordered ﬁeld, 264

ordinary differential equation, 386

orthogonal, 191

orthogonal complement, 203, 521

orthogonal projection, 196

orthonormal, 91, 191

622 Index

orthonormal basis, 91, 191

oscillation, 176, 294

p-adic metric, 247

p-adic numbers, 266

parallelogram law, 91

Parseval’s Theorem, 200, 468

partial differential equation, 386

particular solution, 412

partition of an interval, 154

path, 134, 259

path connected set, 259

Peano’s Theorem, 420

penalty method, 608

perfect set, 104, 107, 354, 364

period, 349

period doubling, 356

periodic function, 126, 132

periodic point, 349

Perturbation Theorem, 416

phase portrait, 351

Picard, 401, 405

piecewise C

, 481

piecewise continuous, 115

piecewise linear function, 314, 541, 546

piecewise Lipschitz, 481

Pigeonhole Principle, 20, 352

Plouffe, 32

points, 88

pointwise convergence, 213

Poisson formula, 436

Poisson kernel, 436, 478

Poisson’s Theorem, 439, 481, 492

polar cone, 597

polar coordinates, 116

polyhedral cone, 600

polyhedral set, 576

polytope, 565

positive kernel function, 508

positively homogeneous function, 586, 589

power series, 226, 232, 277

power set, 9

Principal Axis Theorem, 188

projection, 196, 569

ambiguous meaning of, 569

Projection Theorem, 196

propagation of singularities, 452

proper subset, 5

properly separates, 573

Pythagorean formula, 91

Q, 31

quadratic convergence, 345

quadratic programming, 612

quantiﬁers, 3

quasi-convex function, 587

R, 35

, 559

radius of convergence, 233

Radon’s Theorem, 563

range, 7

rank, 16

Ratio Test, 75

rational function, 122

rational numbers, 31

real numbers, 31, 34

uniqueness of, 264

real part, 455

rearrangement, 81

Rearrangement Theorem, 83

recursion, 24, 52, 53, 63, 65, 126, 252

recursion formula, 170

reﬁnement of a partition, 154

regular curve, 394

relation, 25

relative boundary, 564

relative complement, 6

relative interior, 564

relatively open, 120

Remes’s algorithm, 306

removable singularity, 114

reparametrization, 394

repelling ﬁxed point, 332

repelling periodic point, 349

residual set, 254

Riemann integrable, 156

vector-valued functions, 392

Riemann integral, 153, 156, 474

Riemann sum, 154

Riemann’s Condition, 156

Riemann–Lebesgue Lemma, 474

Riesz basis, 543, 548

Riesz multiresolution, 553

right-differentiable, 146

Rolle’s Theorem, 148

Root Test, 72

Russell’s Paradox, 64

saddlepoint, 604

sawtooth function, 113

scalar multiplication, 12

scaling function, 521, 553

scaling relation, 524

Schroeder–Bernstein Theorem, 63

Schwarz, 188