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Linearization Constraints

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Linearization constraints refer to the limitations imposed on a nonlinear system to approximate its behavior using linear models. These constraints are essential in optimization and control theory, ensuring that the linearized model accurately reflects the dynamics of the original nonlinear system within a specified range of operation.
lightbulbAbout this topic
Linearization constraints refer to the limitations imposed on a nonlinear system to approximate its behavior using linear models. These constraints are essential in optimization and control theory, ensuring that the linearized model accurately reflects the dynamics of the original nonlinear system within a specified range of operation.

Key research themes

1. What transformation and linearization techniques exist to reduce computational complexity in nonlinear optimization models?

This theme centers on the development and evaluation of methods that convert nonlinear optimization problems—often marked by increased computational burden—into equivalent or approximated linear formulations. Such transformations and approximations aim to maintain solution quality while enabling the application of efficient linear programming solvers and reducing computational effort. Key techniques include exact transformations, piecewise linear approximations, Taylor expansions, and parametric linearization.

Key finding: This comprehensive review distinguishes two main classes of linearization approaches: (i) exact transformations that replace nonlinear terms with equivalent linear constraints leading to valid inequalities, and (ii) linear... Read more
Key finding: Introduces the Parametric Linearization Approach (P.L.A) which transforms zero-one nonlinear programming problems into a sequence of linear programming subproblems. By moving linear expansion points (moving linear Taylor... Read more
Key finding: Proposes a piecewise linear approximation methodology for both smooth and nonsmooth nonlinear functions appearing in constrained optimization problems. The approach defines a novel Global Weak Differentiation concept in... Read more
Key finding: Presents a two-phase successive linearization algorithm that solves nonlinear programming problems by generating a sequence of linear programming problems whose solutions progressively satisfy nonlinear constraints with... Read more
Key finding: Evaluates several stochastic linearization methods based on different optimality criteria that minimize various probabilistic functionals, such as the mean square deviation between nonlinear and linear restoring forces or... Read more

2. How can nonlinear system models be standardized or linearized to facilitate the analysis and control across multiple equilibria or nonlinear regimes?

This theme explores methodologies extending classical linearization about equilibria to broader nonlinear regimes or multiequilibrium systems, enabling the use of linear system theory and nonlinear series expansions (e.g., Volterra series) for analysis and control. It includes approaches to nonlinear model standardization, recursive algorithms for transforming nonlinear systems into analyzable forms, and hybrid piecewise linear modeling to capture nonlinear dynamics effectively.

Key finding: Proposes nonlinear model standardization, a generalization of linearization suited for nonlinear systems stable around multiple equilibria and operating beyond infinitesimal neighborhoods. It provides a systematic... Read more
Key finding: Compares the traditional Jacobian linearization method with the Optimal Linear Model approach, which builds local linear models optimal in both state and control terms at arbitrary operating points rather than only... Read more
Key finding: Reviews various approximate linearization methods via state or output feedback that generalize exact differential geometric linearization conditions. Highlights partial linearization, linearization-oriented modeling,... Read more
Key finding: Extends constant rank constraint qualification (CRCQ) concepts from NLP to nonlinear semidefinite programming, presenting sequential variants that incorporate perturbations guaranteeing robustness in iterative algorithms... Read more
Key finding: Introduces weaker variants of the classical nondegeneracy constraint qualification by leveraging the structure of the constraint matrix's eigenvectors and sparsity patterns. The weak-nondegeneracy facilitates global... Read more

3. What role do constraint qualifications and second-order optimality conditions play in ensuring correctness and convergence of linearized or approximated nonlinear optimization solutions?

This theme examines how various constraint qualifications (CQs) and second-order optimality conditions underpin the theoretical validity of linearizations and approximations used in nonlinear optimization. It focuses on recent developments in CQ notions—such as weak-CQs, sequential CQs, and their extensions to conic and semidefinite programs—and their implications for deriving optimality conditions that depend on single Lagrange multipliers, crucial for algorithm convergence and solution correctness.

Key finding: Characterizes the weakest second-order constraint qualifications guaranteeing that second-order optimality conditions relying on a single Lagrange multiplier hold at local minimizers. Introduces new, weaker second-order CQs... Read more
Key finding: Develops sequential optimality conditions—extensions of approximate KKT conditions—to nonlinear conic programming encompassing NLP, semidefinite, and second-order cone programming. Demonstrates that feasible limit points of... Read more
Key finding: Theoretically connects augmented Lagrangian methods to the proximal point algorithm through local duality and saddle-point conditions, providing new insights into step-size selection and demonstrating local linear convergence... Read more
Key finding: Derives explicit equations of motion for general nonlinear constrained mechanical systems even when mass matrices are singular or positive semi-definite, by introducing unconstrained auxiliary systems with positive definite... Read more
Key finding: Identifies two main pitfalls: first, linearizing constraints before deriving optimality conditions leads to incorrect first-order conditions and misleading welfare implications; second, even when linearization of optimality... Read more

All papers in Linearization Constraints

Большое изменение начинается с малого: Прономинальные клитики в тексте древнерусских летописей XII-XV. вв. Аннотация В противоположность западно-и южнославянским языкам, современные восточнославянские языки не имеют энклитических форм... more
This paper explores the use of bound forms in coordination constructions and ʔijjā and ʔijja in Modern Standard Arabic (MSA) and Jordanian Arabic (JA), respectively. Using the Minimalist Program (Chomsky 1995, 2000, 2005) as a theoretical... more
The paper discusses syntactic features of Slavic possessive clitics and Slavic constructions with socalled-Possessor Raising‖. I prove that only a minority of Slavic languages have true phrase-level (NP-level or DP-level) possessive... more
This paper explores the use of bound forms in coordination constructions and ʔijjā and ʔijja in Modern Standard Arabic (MSA) and Jordanian Arabic (JA), respectively. Using the Minimalist Program (Chomsky 1995, 2000, 2005) as a theoretical... more
Clitic climbing, i.e. the realization of one or more clitics in a syntactic constituent hierarchically higher than the clitics' licensing predicate, has been accounted for in terms of a restructuring approach. The embedded infinitive the... more
Andrej A. Zaliznjak's recent book, Drevnerusskie ènklitiki, explains the need for a book about Old Russian enclitics by stating that while the enclitics of the South and West Slavic languages have attracted great interest, those of Old... more
This paper explores the use of bound forms in coordination constructions and ʔijjā and ʔijja in Modern Standard Arabic (MSA) and Jordanian Arabic (JA), respectively. Using the Minimalist Program (Chomsky 1995, 2000, 2005) as a theoretical... more
In this article, we trace the diachronic phases of so-called genitive-dative syncretism in Old Bulgarian, a phenomenon which marks the beginning of the process of disintegration of the Case system in the history of Bulgarian. We base our... more
The paper discusses issues in the grammar of Case marking in the DP by focusing on two interrelated puzzles in the syntax of Bulgarian nominalizations. The first puzzle concerns the ban on strong pronouns to act as DP-internal subjects.... more
This paper presents a novel framework for selecting socially responsible investment (SRI) portfolios. The Hedonic Price Method (HPM) is applied to obtain an evaluation of SRI criteria that is integrated into a multiobjective mathematical... more
This paper gives an account of inalienable possession (IP) in Brazilian Portuguese (BP) that involves semantics and pragmatics, in addition to the expected morpho-syntactic factors. It argues that colloquial BPIP has moved away from the... more
This paper explores the use of bound forms in coordination constructions and ʔijjā and ʔijja in Modern Standard Arabic (MSA) and Jordanian Arabic (JA), respectively. Using the Minimalist Program (Chomsky 1995, 2000, 2005) as a theoretical... more
Th e authors are interested in the semantic (= not formal) evolution of the dative case relationship. Th ey carry their analysis in the framework of the anthropocentric case theory and argue that the referent of the dative NP (NP D) is... more
This paper analyses the placement of clitics that occupy the so-called "second" position in Serbian, in which both the first word or the first constituent can serve as host positions for clitics. In both corpus investigations... more
This paper deals with a small set of data from clusters of three clitics in Spanish that questions the empirical adequacy and scope of previous analyses of clitic clusters in Romance. It is shown that the output of the Spurious se Rule is... more
I discuss the methodology and conclusiveness of the corpus-based historical linguistics and analyze two formal models predicting the language-internal variation in Early Old Russian syntax. Linguistic models claiming a rigid distribution... more
Apologies for the duplication. This 2012 paper of mine was automatically recognized by academia.edu. A slightly edited version of this text is available as a chapter of my 2021 book А.В.Циммерлинг. От интегрального к аспективному.... more
In theory, investors choosing to invest only in socially responsible entities restrict their investment universe and should thus be penalized in a mean-variance framework. When computed, this penalty is usually viewed as valid for all... more
Further steps Can we get rid of such cases? Could they simply be mistakes? No! Too frequent, too systematic… Are they metri causa? Maybe... But: "preventive censure" (Jakobson & Bogatyrev 1929) → inevitable premise: All attested data are... more
Apologies for cross-posting. This 2011 paper of mine published in Issue 10 (17) in the "Computational Linguistics and Intellectual Technologies" was automatically recognized by academia.edu. Статья содержит классификацию разновидностей... more
This paper explores the use of bound forms in coordination constructions and ʔijjā and ʔijja in Modern Standard Arabic (MSA) and Jordanian Arabic (JA), respectively. Using the Minimalist Program (Chomsky 1995, 2000, 2005) as a theoretical... more
This paper is addressed the issues in Slavic clitic syntax, with focus on clausal-initial proclitic-enclitic sequences in Old Russian. • I argue that Early Old Russian in XI-XII centuries had a parameter setting licensing free... more
The paper discusses syntactic features of Slavic possessive clitics and Slavic constructions with socalled-Possessor Raising‖. I prove that only a minority of Slavic languages have true phrase-level (NP-level or DP-level) possessive... more
This paper accounts for the distribution of two second position effects, the V2 (verb second) order observed in continental Germanic languages and second position cliticization, attested in some Slavic languages. It shows that it is... more
In this article, we trace the diachronic phases of so-called genitive-dative syncretism in Old Bulgarian, a phenomenon which marks the beginning of the process of disintegration of the Case system in the history of Bulgarian. We base our... more
This paper discusses clitic reduplication constructions in Bulgarian. In contrast to traditional analyses, it distinguishes clitic doubling proper, which is restricted to clauses with psych and physical perception predicates, from other... more
The study offers novel evidence on the grammar and processing of clitic placement in heritage languages. Building on earlier findings of divergent clitic placement in heritage European Portuguese and Serbian, this study extends this line... more
The authors are interested in the semantic (= not formal) evolution of the dative case relationship. They carry their analysis in the framework of the anthropocentric case theory and argue that the referent of the dative NP (NPD) is the... more
The world's languages with clitic clusters pattern with four major types of the word order systems, depending on the presence or absence of the 2P condition and the parameter of clitic-verb adjacency. Bulgarian has a double-focus system... more
The paper deals with the acquisition of Slovak word order in written texts of students of Slovak as a foreign language. Its attention is focused on identifying the correct and incorrect placement of enclitic components, and their... more
Andrej A. Zaliznjak's recent book, Drevnerusskie ènklitiki, explains the need for a book about Old Russian enclitics by stating that while the enclitics of the South and West Slavic languages have attracted great interest, those of Old... more
This paper presents an analysis of certain aspects of Czech sentential clitics in Higher Order Grammar. I focus on the relative order of clitics within the clitic cluster. The overall aim of the paper is to show that constraints governing... more
This paper investigates the issue of head directionality in Old Slavic. This issue has played an important role in diachronic studies on Germanic, in which a switch in head directionality was assumed to have triggered word order changes... more
This dissertation discusses the non-agentive Bulgarian verbal construction opitva se. Though normally it carries the meaning of ‘to try’, this special usage allows a meaning roughly equivalent to French risquer (‘Il risque de pleuvoir’)... more
The paper deals with the word order of reflexive sě, which is an item on the boundary between a pronominal form and a discrete morpheme. In the first part of the study, we investigate the (en)clitic status of sě in eight books of the... more
The paper deals with the word order of reflexive sě, which is an item on the boundary between a pronominal form and a discrete morpheme. In the first part of the study, we investigate the (en)clitic status of sě in eight books of the... more
Pronominal clitics comprise one of the important traits of the majority of West Iranian languages. Nevertheless, while these person clitics have been the subject of virtually systematic studies in certain languages, e.g. Central Kurdish... more
Andrej A. Zaliznjak's recent book, Drevnerusskie ènklitiki, explains the need for a book about Old Russian enclitics by stating that while the enclitics of the South and West Slavic languages have attracted great interest, those of Old... more
The authors are interested in the semantic (= not formal) evolution of the dative case relationship. They carry their analysis in the framework of the anthropocentric case theory and argue that the referent of the dative NP (NPD) is the... more
Th e authors are interested in the semantic (= not formal) evolution of the dative case relationship. Th ey carry their analysis in the framework of the anthropocentric case theory and argue that the referent of the dative NP (NP D) is... more
Purpose – The purpose of this paper is first to give an in-depth discussion of the criticism of socially responsible investment's (SRI) alleged incompatibility with the concept of rational investment constituting an inferiority to... more
We present a framework for inverse optimization in a Markowitz portfolio model that is extended to include a third criterion. The third criterion causes the traditional nondominated frontier to become a surface. Until recently, it had not... more
This paper investigates whether V-toT movement is attested in Old Russian, by examining word order between verbs and other elements such as overt subjects and low adverbs. The position of the be-auxiliary is also examined as a diagnostics... more
This paper discusses clitic reduplication constructions in Bulgarian. In contrast to traditional analyses, it distinguishes clitic doubling proper, which is restricted to clauses with psych and physical perception predicates, from other... more
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