Variational Analysis and Generalized Differentiation II: Applications

5 Constrained Optimization and Equilibria . . . . . . . . . . . . . . . . . . 3

5.1 Necessary Conditions in Mathematical Programming . . . . . . . . . 3

5.1.1 Minimization Problems with Geometric Constraints . . . 4

5.1.2 Necessary Conditions under Operator Constraints . . . . . 9

5.1.3 Necessary Conditions under Functional Constraints . . . . 22

5.1.4 Suboptimality Conditions for Constrained Problems . . . 41

5.2 Mathematical Programs with Equilibrium Constraints . . . . . . . 46

5.2.1 Necessary Conditions for Abstract MPECs . . . . . . . . . . . 47

5.2.2 Variational Systems as Equilibrium Constraints . . . . . . . 51

5.2.3 Refined Lower Subdifferential Conditions

for MPECs via Exact Penalization . . . . . . . . . . . . . . . . . . . 61

5.3 Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3.1 Optimal Solutions to Multiobjective Problems . . . . . . . . 70

5.3.2 Generalized Order Optimality . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.3 Extremal Principle for Set-Valued Mappings . . . . . . . . . . 83

5.3.4 Optimality Conditions with Respect

to Closed Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3.5 Multiobjective Optimization

with Equilibrium Constraints . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Subextremality and Suboptimality at Linear Rate . . . . . . . . . . . 109

5.4.1 Linear Subextremality of Set Systems . . . . . . . . . . . . . . . . 110

5.4.2 Linear Suboptimality in Multiobjective Optimization . . 115

5.4.3 Linear Suboptimality for Minimization Problems . . . . . . 125

5.5 Commentary to Chap. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

XX Contents

6 Optimal Control of Evolution Systems in Banach Spaces . . 159

6.1 Optimal Control of Discrete-Time and Continuoustime

Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.1.1 Differential Inclusions and Their Discrete

Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.1.2 Bolza Problem for Differential Inclusions

and Relaxation Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.1.3 Well-Posed Discrete Approximations

of the Bolza Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.1.4 Necessary Optimality Conditions for Discrete-

Time Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.1.5 Euler-Lagrange Conditions for Relaxed Minimizers . . . . 198

6.2 Necessary Optimality Conditions for Differential Inclusions

without Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.2.1 Euler-Lagrange and Maximum Conditions

for Intermediate Local Minimizers . . . . . . . . . . . . . . . . . . . 211

6.2.2 Discussion and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.3 Maximum Principle for Continuous-Time Systems

with Smooth Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.3.1 Formulation and Discussion of Main Results . . . . . . . . . . 228

6.3.2 Maximum Principle for Free-Endpoint Problems . . . . . . . 234

6.3.3 Transversality Conditions for Problems

with Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 239

6.3.4 Transversality Conditions for Problems

with Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 244

6.4 Approximate Maximum Principle in Optimal Control . . . . . . . . 248

6.4.1 Exact and Approximate Maximum Principles

for Discrete-Time Control Systems . . . . . . . . . . . . . . . . . . 248

6.4.2 Uniformly Upper Subdifferentiable Functions . . . . . . . . . 254

6.4.3 Approximate Maximum Principle

for Free-Endpoint Control Systems . . . . . . . . . . . . . . . . . . 258

6.4.4 Approximate Maximum Principle under Endpoint

Constraints: Positive and Negative Statements . . . . . . . . 268

6.4.5 Approximate Maximum Principle

under Endpoint Constraints: Proofs and

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

6.4.6 Control Systems with Delays and of Neutral Type . . . . . 290

6.5 Commentary to Chap. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

7 Optimal Control of Distributed Systems . . . . . . . . . . . . . . . . . . . 335

7.1 Optimization of Differential-Algebraic Inclusions with Delays . . 336

7.1.1 Discrete Approximations of Differential-Algebraic

Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

7.1.2 Strong Convergence of Discrete Approximations . . . . . . . 346

Contents XXI

7.1.3 Necessary Optimality Conditions

for Difference-Algebraic Systems . . . . . . . . . . . . . . . . . . . . 352

7.1.4 Euler-Lagrange and Hamiltonian Conditions

for Differential-Algebraic Systems . . . . . . . . . . . . . . . . . . . 357

7.2 Neumann Boundary Control

of Semilinear Constrained Hyperbolic Equations . . . . . . . . . . . . . 364

7.2.1 Problem Formulation and Necessary Optimality

Conditions for Neumann Boundary Controls . . . . . . . . . . 365

7.2.2 Analysis of State and Adjoint Systems

in the Neumann Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 369

7.2.3 Needle-Type Variations and Increment Formula . . . . . . . 376

7.2.4 Proof of Necessary Optimality Conditions . . . . . . . . . . . . 380

7.3 Dirichlet Boundary Control

of Linear Constrained Hyperbolic Equations . . . . . . . . . . . . . . . . 386

7.3.1 Problem Formulation and Main Results

for Dirichlet Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

7.3.2 Existence of Dirichlet Optimal Controls . . . . . . . . . . . . . . 390

7.3.3 Adjoint System in the Dirichlet Problem . . . . . . . . . . . . . 391

7.3.4 Proof of Optimality Conditions . . . . . . . . . . . . . . . . . . . . . 395

7.4 Minimax Control of Parabolic Systems

with Pointwise State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 398

7.4.1 Problem Formulation and Splitting . . . . . . . . . . . . . . . . . . 400

7.4.2 Properties of Mild Solutions

and Minimax Existence Theorem . . . . . . . . . . . . . . . . . . . . 404

7.4.3 Suboptimality Conditions for Worst Perturbations . . . . . 410

7.4.4 Suboptimal Controls under Worst Perturbations . . . . . . . 422

7.4.5 Necessary Optimality Conditions

under State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

7.5 Commentary to Chap. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

8 Applications to Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

8.1 Models of Welfare Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

8.1.1 Basic Concepts and Model Description . . . . . . . . . . . . . . . 462

8.1.2 Net Demand Qualification Conditions for Pareto

and Weak Pareto Optimal Allocations . . . . . . . . . . . . . . . 465

8.2 Second Welfare Theorem for Nonconvex Economies . . . . . . . . . . 468

8.2.1 Approximate Versions of Second Welfare Theorem . . . . . 469

8.2.2 Exact Versions of Second Welfare Theorem . . . . . . . . . . . 474

8.3 Nonconvex Economies with Ordered Commodity Spaces . . . . . . 477

8.3.1 Positive Marginal Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

8.3.2 Enhanced Results for Strong Pareto Optimality . . . . . . . 479

8.4 Abstract Versions and Further Extensions . . . . . . . . . . . . . . . . . . 484

8.4.1 Abstract Versions of Second Welfare Theorem . . . . . . . . . 484

8.4.2 Public Goods and Restriction on Exchange . . . . . . . . . . . 490

8.5 Commentary to Chap. 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

XXII Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

List of Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Glossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599