Optimality Conditions and Shadow Prices *

2020 ◽  
pp. 49-63
Author(s):  
Henry Wolkowicz
Keyword(s):  
Optimality Conditions ◽  
Shadow Prices

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Analysis and design of thermo-mechanical interfaces

2012 ◽  
Vol 60 (2) ◽  
pp. 205-213
Author(s):  
K. Dems ◽  
Z. Mróz
Keyword(s):  
Optimality Conditions ◽  
Mechanical Loading ◽  
Material Properties ◽  
Structural Response ◽  
External Boundary ◽  
Mechanical Loads ◽  
Internal Interface ◽  
Analysis And Design ◽  
First Order ◽  
Mechanical Nature

Abstract. An elastic structure subjected to thermal and mechanical loading with prescribed external boundary and varying internal interface is considered. The different thermal and mechanical nature of this interface is discussed, since the interface form and its properties affect strongly the structural response. The first-order sensitivities of an arbitrary thermal and mechanical behavioral functional with respect to shape and material properties of the interface are derived using the direct or adjoint approaches. Next the relevant optimality conditions are formulated. Some examples illustrate the applicability of proposed approach to control the structural response due to applied thermal and mechanical loads.

Comments on "Shadow Pricing and Macroeconomic Analysis: Some Illustrations from Pakistan" by Squire et al.

1979 ◽  
Vol 18 (2) ◽  
pp. 113-115
Author(s):  
T. N. Srinivasan
Keyword(s):  
General Equilibrium ◽  
Equilibrium Model ◽  
General Equilibrium Model ◽  
Equilibrium Analysis ◽  
Partial Equilibrium ◽  
Shadow Prices ◽  
Essential Point ◽  
Economic Issues ◽  
Shadow Pricing ◽  
Extreme Sensitivity

The paper is too long for conveying the message that shadow pricing used as a method of analysis in micro-economic issues of project selection is also useful for analysing macro-economic issues, such as foreign and domestic borrowing by the government, emigration, etc. Much of the methodological discussion in the paper is available in a readily accessible form in several publications of each of the coauthors; In contrast, the specific application of the methodology to Pakistani problems is much too cavalier. While it is hard to disagree with the authors' claim that shadow pricing "constitutes a relatively informal attempt to capture general equilibrium effects" (p. 89, emphasis added), their depiction of traditional analysis is a bit of a caricature: essentially it sets up a strawman to knock down. After all in the traditional partial equilibrium analysis, the caveat is always entered that the results are possibly sensitive to violation of the ceteris paribus assumptions of the analysis, though often the analysts will claim that extreme sensitivity is unlikely. Analogously, the shadow pricing method presumes "stationarity" of shadow prices in the sense that they are “independent of policy changes under review" (p. 90). The essential point to be noted is that the validity of this assertion or of the "not too extreme sensitivity" assertion of partial equilibrium analysts can be tested only with a full scale general equilibrium model! At any rate this reviewer would not pose the issue as one of traditional partial equilibrium macro-analysis versus shadow pricing as an approximate general equilibrium analysis, but would prefer a description of project analysis as an approach in which a macro-general equilibrium model of a manageable size (implicit or explicit) is used to derive a set of key shadow prices which are then used in a detailed micro-analysis of projects.

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