Stochastic Modeling and Optimization Methods for Critical Infrastructure Protection, Volume 1 E-Book

Alexei A. Gaivoronski

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: ISTE Invoiced
• Sprache: Englisch

Seitenzahl: 404

Veröffentlichungsjahr: 2025

Beschreibung

Stochastic Modeling and Optimization Methods for Critical Infrastructure Protection is a thorough exploration of mathematical models and tools that are designed to strengthen critical infrastructures against threats – both natural and adversarial. Divided into two volumes, this first volume examines stochastic modeling across key economic sectors and their interconnections, while the second volume focuses on advanced mathematical methods for enhancing infrastructure protection.

The book covers a range of themes, including risk assessment techniques that account for systemic interdependencies within modern technospheres, the dynamics of uncertainty, instability and system vulnerabilities. The book also presents other topics such as cryptographic information protection and Shannon's theory of secret systems, alongside solutions arising from optimization, game theory and machine learning approaches.

Featuring research from international collaborations, this book covers both theory and applications, offering vital insights for advanced risk management curricula. It is intended not only for researchers, but also educators and professionals in infrastructure protection and stochastic optimization.

Leseprobe

Series EditorNikolaos Limnios

Stochastic Modeling and Optimization Methods for Critical Infrastructure Protection 1

Stochastic Modeling

Edited by

Alexei A. Gaivoronski

Pavel S. Knopov

Vladimir I. Norkin

Volodymyr A. Zaslavskyi

First published 2025 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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© ISTE Ltd 2025The rights of Alexei A. Gaivoronski, Pavel S. Knopov, Vladimir I. Norkin and Volodymyr A. Zaslavskyi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.

Library of Congress Control Number: 2025931326

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-83669-027-6

Preface

The book presents mathematical models and tools for enhancing critical infrastructure protection against natural threats and adversarial attacks. The book consists of two volumes; the first is devoted to stochastic modeling of various critical sectors of economics and their interconnections, and the second considers mathematical tools for enhancing critical infrastructure protection.

These two volumes offer new results on risk assessment methods, taking into account the large number of systemic connections between the structures of the modern technosphere and the graded nature of damage probability from natural and human-made disasters. The cumulative effect from various risk sources will be investigated, which becomes an important generator of uncertainty and instability due to the nonlinear increase in system vulnerability and a decrease in its adaptation capabilities, when small impacts cause unstable system behavior. The generalizations of the attack and defense antagonistic game, and the rational distribution of defense resources are considered, and methods for solving the corresponding optimization problems are proposed. New mathematical methods of machine learning, based on quasi-gradient methods of stochastic programming, are reviewed.

The research is based on the authors’ results on the theoretical foundations of stochastic systems with aftereffect, stochastic optimal control theory, robust methods of non-smooth and stochastic optimization for solving special optimization problems with a large number of variables and constraints, methods of multi-criteria dynamic optimization, methods of solving two-level stochastic programming problems, stochastic minimax game problems, and graph theory optimization problems. Basic concepts, methods and directions of cryptographic protection of information are considered. The working principles of classical cryptography ciphers and the main results of Shannon’s theory of secret systems are presented. The obtained theoretical results will be applied to solving important problems of risk assessment and monitoring the state of critical infrastructure objects.

The chapters are the result of a series of international projects involving leading scientists from various research institutes in Austria, Norway, Ukraine and other countries in the field of modern stochastic optimization and decision making. These projects were aimed at developing a curriculum for risk management at the master’s and doctoral levels. The studies included in this book contain new research results and also take into account the requirements of research-based education on these topics.

Acknowledgments

Work on parts of the book was supported by the grant CPEA-LT-2016/10003 “Advanced collaborative program for research based education on risk management in industry and services under global economic, technological and environmental changes: enhanced edition” of Norwegian Directorate for Higher Education and Skills (HKDIR), managed by the Norwegian University of Science and Technology (NTNU), Trondheim, Norway. It was also supported by the project 2020.02/0121 “Analytical methods and machine learning in control theory and decision-making in conditions of conflict and uncertainty” funded by the National Research Foundation of Ukraine. The work was also supported by the joint project between the International Institute for Applied Systems Analysis (IIASA) and the National Academy of Sciences of Ukraine (NASU) on “Integrated robust modeling and management of food-energy-water-land use nexus for sustainable development” (22-501) and by the projects 0122U00052 and 2.3/25-P of the National Academy of Sciences of Ukraine.

Alexei A. GAIVORONSKIPavel S. KNOPOVVladimir I. NORKINVolodymyr A. ZASLAVSKYI

March 2025

PART 1Model-Based Risk Management in Critical Economic Sectors

Introduction to Part 1

Integrated solutions and procedures for combining distributed models for security modeling and management in the food, energy, water and environment sectors are proposed. The links between two-stage stochastic optimization and machine learning problems are discussed, and a new model using quantile regression for assessing the impact of land use practices on the environment is proposed. Also, new mathematical models for studying the vulnerabilities of critical infrastructure are proposed.

A multi-stage, multi-horizon stochastic equilibrium model for analyzing energy markets is studied, with particular attention applied to infrastructure development and renewable energy policy in perfect and imperfect market structures.

This section also proposes new approaches to managing energy systems – important objects of critical infrastructure. The task of optimizing the structure of the energy production portfolio is solved in order to combine various energy production technologies to maximize the value of the portfolio, ensure its safety and prevent the occurrence of critical situations.

Note

Introduction written by Alexei A. GAIVORONSKI and Pavel S. KNOPOV.

1Integrated Solutions and Distributed Models’ Linkage Procedures for Food–Energy–Water–Environmental Nexus Security Modeling and Management

This chapter discusses a new modeling approach enabling the linkage of distributed individual food, energy, water optimization models under joint (e.g. water and land) resource constraints, uncertainty and asymmetric information (ASI). The approach is based on an iterative stochastic quasigradient (SQG) solution procedure of, in general, non-smooth, non-differentiable optimization. The SQG procedure organizes an iterative computerized negotiation between individual food–energy–water (FEW) systems (models) representing intelligent agents. The convergence of the procedure to the socially optimal solution is based on the results of non-differentiable optimization providing a new type of machine learning algorithm. The linkage problem can be viewed as a general endogenous reinforced learning problem. The models act as “agents” that communicate with a “central hub” (a regulator) and take decisions in order to maximize the “cumulative reward”. In this way, they continue to be the same individual models and different modeling teams do not need to exchange information about their models – instead, they only need to harmonize the inputs and outputs that are part of the joint resource constraints. In this way, the agents link their models into an integrated model (IM) under ASI.

The convergence of the solution of the linked models to the solution of the hard-IM is discussed. Application of the approach is illustrated with a case study linking distributed agricultural, water and energy sector models. This approach can be effectively used for decentralized, deregulated planning of interdependent agricultural, energy and water systems.

1.1. Introduction

The increasing interdependencies among food–energy–water–environmental (FEWE) sectors require integrated coherent planning and coordinated policies for sustainable development and nexus security. The sectors become more interconnected because they utilize common, often rather limited, resources, both natural (e.g. land, water and air quality) and socioeconomic (e.g. investments and labor force). For example, land and water are needed not only for agricultural production but also for hydropower generation, coal mining and processing, power plants cooling, renewable energy and hydrogen production.

Detailed sectoral and regional models have traditionally been used to anticipate and plan desirable developments of respective sectors and regions. These models operate with a set of feasible decisions and aim to select a solution optimizing a sector- or region-specific objective function. When interdependencies between sectors and regions are increasing, such an independent analysis without accounting for interconnections can become misleading and even dangerous. Hence, it is necessary to link the sectoral (regional) models together to derive truly integrated solutions. Interdependent FEWE security goals contribute immensely to signifying the nexus security between sectors and regions (Zagorodny et al. 2020, 2024; Ermolieva et al. 2021a, 2021b).

In this chapter, we consider the problem of linking individual sectoral and/or regional linear programming (LP) models into a cross-sectoral IM in the presence of joint constraints, when “private” information about the models is not available or it cannot be shared by modeling teams (sectoral agencies), that is, under ASI. This approach provides a means of decentralized cross-sectoral coordination and enables us to investigate policies in interdependent systems in a “decentralized” fashion. This enables more stable and resilient systems’ performance and resource allocation as compared to the independent policies designed by separate models without accounting for interdependencies.

Limited resources can be allocated between systems (sectors/regions) in many ways. For example, Böhringer and Rutherford (2009) consider integrating mathematical programming models of the energy system into a general equilibrium (GE) model of the overall economy. Unfortunately, the convergence of the iterative procedure integrating the models could not be proven. For resource and production allocation with the generalized Nash equilibrium (GNE) approach, the existence, uniqueness and stability of the GNE, and a realistic large-scale implementation of this concept cannot be guaranteed. Ermoliev and von Winterfeldt (2012) demonstrate the complexity of the game-theoretic approaches, for example, the Stackelberg leadership model, emerging due to quite unrealistic assumptions that each player (sector/region) possesses knowledge about other players.

Traditional integrated deterministic optimization modeling incorporates goals, constraints and data of all models into a single code (hard integration), which can be considered as a multi-criteria optimization problem (Ogryczak 2000). In the case of separate distributed models and ASI, the linkage requires (see sections 1.2.2–1.2.4) specific non-smooth optimization methods. Problems under ASI are addressed with the agency theory (Gaivoronski and Werner 2012), in particular, on how to motivate information exchange. In this chapter, we consider, in a sense, the opposite and minimize the necessity to exchange information.

Our approach for linking separate optimization models under ASI is based on the parallel solving of equivalent non-smooth optimization models by an iterative stochastic quasigradient (SQG) procedure (Ermoliev 1976, 2009), based on subgradients or generalized gradients (Ermoliev 1976; Rockafeller 1981; Ermoliev and Norkin 1997) converging to an optimal welfare maximizing linkage solution, that is, to the solution of a “hard-integrated” model (Ermoliev et al. 2022). This approach does not require us to share details about models’ specification. We can assume that there is a network of distributed computers connecting individual computer models with the computer of a “social planner” (decision-makers or regulatory agencies), who attempts to achieve the best result for all sectors/regions (parties) involved. The linkage procedure can be interpreted as a kind of a “decentralized market system” (Ermoliev et al. 2015, and references therein). According to this procedure, the optimization of sectoral/regional goals under individual constraints is performed independently and in parallel, without considering joint constraints. Joint constraints can be imposed on total production, natural and financial resource use, emissions, pollution and joint FEWE security targets. The constraints can establish supply–demand relationships between the systems enabling us to estimate optimal production, resource use and emission quotas for each system. The balance between the total energy (including biofuels) production and demand defines energy security, agricultural production and consumption reflect food security, and total emissions and pollution constraints correspond to environmental security. The joint FEWE constraints satisfaction establishes the FEWE security nexus (Zagorodny and Ermoliev 2013; Ermolieva et al. 2016, 2021a; Zagorodny et al. 2020, 2024). After the independent optimization using initial approximations of various (e.g. production, resource use and emission) quotas, the sectors/regions provide a social planner with the information on their actual production, resource use and respective shadow prices. The planner checks if the joint constraints are fulfilled. If not, that is, there is “excess demand” or “excess supply” (i.e. total resource use, production and emissions by all systems are higher/lower than required), the planner revises the individual systems’ quotas via shifting their current approximation in the direction defined by the corresponding dual variables. Thus, shadow prices signal systems to adjust their activities accordingly. Formally, the procedure is described in sections 1.2.3–1.2.5.

In this way, the linkage allows us to avoid “hard linking” of the models in a single code, which is not possible because the systems (agencies) may not want to share the information or because the individual models are too detailed and complex to be “hard-linked”. The approach saves reprogramming efforts and enables parallel distributed (decentralized) computations of sectoral models instead of a large-scale integrated (centralized) model. This also preserves the original models in their initial state for other linkages. The use of detailed sectoral and regional models instead of their aggregated simplified versions enables us to also account for critically important local details. Similar computerized, decentralized “negotiation” processes between distributed models (agents) have been developed for the design of robust carbon trading markets (e.g. Ermoliev et al. 2015, and references therein). The linkage procedure can be considered as a new machine learning algorithm, namely, as a general endogenous reinforced learning problem of how software agents (models) take decisions in order to maximize the cumulative reward (total welfare) (Ermolieva et al. 2021b)

This chapter is organized as follows. Section 1.2 discusses the problem of models’ linkage under joint constraints. Section 1.2.1 presents a short overview and main shortfalls of several existing approaches. Section 1.2.2 formulates the problem of distributed LP models’ linkage in the presence of joint resource constraints and ASI. Sections 1.2.3–1.2.5 outline the details and the main properties of the linkage solution procedure based on the parallel solving of the equivalent non-smooth optimization model following a simple iterative subgradient algorithm. Section 1.3 illustrates the application of the methodology to link detailed energy and agricultural production planning models under joint constraints on water and land use. In addition, the joint constraints can impose restrictions on total energy production by energy sector (electricity, gas, diesel, etc.) and land use sector (biodiesel and methanol), total energy use by energy and agricultural sectors, and total agricultural production by distributed farmers/regions. Section 1.4 concludes and outlines potential further extensions of the approach, for example, to include more details of energy and natural resources dynamics in general.

1.2. Linking distributed optimization models under joint resource constraints

1.2.1. Social equilibrium game

In the absence of coordination between systems (sectors and regions), they can act selfishly and aim at maximizing their own objective function. They can attempt to secure as high resource quotas as possible. Such a situation can be modeled using the non-cooperative game-theoretic framework. For example, social equilibrium games (Harker 1991) have been formulated to include joint constraints. The GNE solution, if it exists, allocates production and resources among systems (sectors/regions) fulfilling the joint constraint. However, the decisions are made independently and collective efforts for managing common resources are ignored. Importantly, the existence, uniqueness and stability of the GNE, as well as a realistic large-scale implementation of this concept, cannot be guaranteed as emphasized by Harker (1991). Moreover, in Harker (1991), it is highlighted that the GNE solutions set is rarely connected. Hence, a complete analysis of equilibriums in this case is a complex task, requiring additional assumptions.

The analysis can become even more complex if the joint constraints are based on the equilibrium (optimality) conditions arising from the problem formulated in the form of a principal–agent game or a leader–follower Stackelberg game (Ermoliev and von Winterfeldt 2012). For example, in the case of non-smooth goal functions required for linking systems under ASI (distributed models’ optimization), the use of optimality conditions would require implicit sets of generalized gradients. Due to the computational complexity, heuristic methods are often used; however, they are lacking a rigorous convergence proof.

Böhringer and Rutherford (2009) discuss linking bottom-up mathematical programming models of the energy system into a top-down GE model of the overall economy. The paper shows that the formulation of market equilibrium conditions by using complementarity equations permits integration of models, but the convergence of the iterative procedure integrating the models cannot be guaranteed. In specific cases, models of GE are reduced to optimization problems (Norkin 1999).

Ermoliev and von Winterfeldt (2012) demonstrate that the complexity of the game-theoretic approaches is due to quite unrealistic assumptions that each player (sector/region) is in the possession of knowledge with regard to exact and unique responses of other players. Therefore, even in the simplest linear cases, this assumption leads to extremely complex discontinuous problems. More realistic assumptions of uncertain response functions in combination with a concept of robust decisions results in stable large-scale solutions.

A vast literature exists regarding important problems and methods for distributed systems’ optimization (Yang et al. 2019) under joint constraints, for example, optimal control and economic dispatch in smart grids (Prakash and Nygard 2017), agricultural production planning for the multi-farmer systems (Alemany et al. 2021), network optimization (Liang et al. 2019) and optimal transportation problems (Dean and Cortés 2015; Galichon 2016). However, these approaches consider optimization of a total objective function representing a sum of individual objective functions of the involved systems. Thus, the problems assume full information is available to a social planner about the systems. They are formulated similarly to the traditional integrated “centralized” optimization modeling combining goals, constraints and data of all models into a single code.

Our problem is more complex as it deals with the coordination of decentralized systems’ models in the presence of joint constraints and ASI. In this case, the approach is based on a specific iterative non-smooth optimization procedure (see sections 1.2.3–1.2.5). As we noted, integrated solution of separate LP models under ASI cannot be accomplished by LP methods. In section 1.2.2, we formulate the problem of distributed systems optimization in the presence of joint resource constraint under ASI, and in sections 1.2.3–1.2.5, we discuss the models’ linkage algorithm and its convergence characteristics.

1.2.2. LP models under joint constraints

The basic problem of distributed individual sectoral (or regional) LP models optimization under joint resource constraint can be formulated as follows.

Assume that there are K systems’ models formulated in the following LP form:

[1.1]

[1.2]

[1.3]

where components of vector x(k) are unknown variables, vector b(k) defines system-specific demand or resource constraints and vector c(k) corresponds to net unit profits, k = 1,2, …, K. The common resource constraint [1.4] connects the systems through a common unknown (linkage) variable y(k)

[1.4]

where y(k) defines resource quota allocated to system k. In this formulation, equation [1.3] defines individual systems’ constraints and formula [1.4] establishes systemic supply–demand relationships among systems by distributing resources y(k). If matrix D(k) defines the marginal resource use by system k and d is the total available resource, d ≥ 0, then the quotas y(k) fulfill the joint resource constraint on the common resources use

[1.5]

Thus, the overall problem is to maximize the individual objective function [1.1] for each system k by choosing x(k) and y(k) from the feasible set defined by [1.2] and [1.3], so that [1.4] and [1.5] are also fulfilled.

In the presence of full information about systems, the problem of models’ linkage can be formulated and solved by a central planner (regulator) as a total net profit maximization

[1.6]

s.t. to constraints [1.2–1.5], k = 1,2,…, K. The net profits can be defined as total net profit of production costs, possible taxes and other expenses.

If the information about parameters b(k), c(k), A(k), B(k), x(k) of systems k is not available to the planner, the integrated LP model [1.2–1.6] under ASI cannot be solved by the LP method due to the lack of common information about submodels.

In this situation, the consistent approach for linking distributed optimization models is based on the parallel solving of an equivalent non-smooth optimization model following a simple iterative subgradient algorithm. The convergence and other properties of the algorithm are presented in section 1.2.4. The proposed linkage approach does not require us to know full information about models’ specification, and it can be seen as an endogenous reinforced learning algorithm describing how distributed agents (models) can take decisions to maximize the “cumulative reward”. Sections 1.2.3–1.2.5 outline various aspects of the algorithm.

1.2.3. Non-smooth model and linking algorithm

A core part of the algorithm is a central hub computer that recalculates the resource quotas y by shifting their current approximation in the direction defined by the corresponding vectors of dual variables (shadow prices of resources) from the primal optimization problems. These quotas are received by sectorial/regional computers enabling parallel computations of solutions and fast adjustments of vector y. Ermoliev (1980) initially introduced the idea of this algorithm and current computer capacities enable its implementation in large-scale models used to support decisions.

Consider the main implicit maximization problem. For a given vector y = (y(1), …, y(K)), let us denote the optimal value of function [1.6] under constraints [1.2–1.4] by F(y); in other words, in this function x(k)(y) are optimal solutions to [1.1] under [1.2–1.4], ignoring joint constraints [1.5]. Therefore,

where f(k)(y) = c(k), x(k)(y)) are concave non-differentiable (continuously) or non-smooth functions.

The following algorithm defines a rule for adjusting y toward an optimal y* that maximizes function F(y) under the joint constraints [1.5] defining the feasible set Y.

Consider an arbitrary feasible solution ys = (ys(1), …,ys(K)) for iteration s = 1,2, … of the algorithm. For given quotas ys = (ys(1), …, ys(K)), independently and in parallel, computers of sectors/regions solve primal models [1.1–1.4] and obtain primal solutions xs(k) = xs(k)(ys) together with the corresponding shadow prices of resources, that is, solutions (us(k), vs(k)) of the dual problems

[1.7]

[1.8]

[1.9]

k = 1,2, …, where vectors vs(k) are the driving force of algorithm [1.10].

The next approximation of quotas ys+1 = (ys+1(1), …, ys+1(K)) is derived by the computer of the central hub by shifting ys in the direction of vector vs = (vs(1), …, vs(K)), that is, optimal dual variables (shadow prices) corresponding to constraints [1.4]. Hence, we have an iterative procedure defining, in a sense, the artificial “intellect” of the designed solution system:

[1.10]

where ρs is an iteration-dependent multiplier, which is a method’s parameter, and πY(·) is the orthogonal projection operator onto set Y (see, for example, Ermoliev 1976, 1980, 2009).

The orthogonal projection ys+1 of vector onto Y is calculated by minimizing the quadratic function ∥ys + ρsvs − y∥2 subject to joint constraints [1.5]. This minimization is very fast due to ρsvs → 0, as vectors vs are bounded optimal dual solutions and if ys is used as an initial approximation for ys+1.

Vector vs defines sub-gradient of the continuously non-differentiable function F(x). This, and the convergence of solutions ys to an optimal solution of the linkage problem [1.2]–[1.6] as s → ∞ is analyzed in section 1.2.4. The step-size ρs is chosen from rather general and natural requirements: ρs ≥ 0, , because generalized gradients are not the increasing directions of functions.

Although the standard sub-gradient projection method converges without condition , the proposed linkage algorithm for problems under ASI [1.10] requires this additional condition to enable the convergence of not only function F(ys) but also solutions ys. This allows us to propose a simple stopping criterion enabling the independent optimization of interdependent sectors by [1.10].

1.2.4. Convergence of the linking algorithm (stopping criterion)

Assume there exist solutions x(k)(y) of all K sectoral/regional models for feasible y satisfying constraints [1.5].

Then:

Functions

f

(

k

)

(

y

) = (

c

(

k

)

,

x

(

k

)

(

y

)), are concave continuously non-differentiable functions for all

k

.

The dual problem

[1.7]

–

[1.9]

has a solution (

u

(

k

)

(

y

),

v

(

k

)

(

y

)) for all

k

, and these solutions satisfy the stopping criterion of the linkage algorithm:

This proposition leads to the following important fact (Ermoliev 1976, 1980, 2009), which is fundamental for solving the linkage problem through maximizing non-differentiable function F(y) by algorithm [1.10]:

For any feasible solution

z

and

y

,

f

(

k

)

(

y

) −

f

(

k

)

(

z

) ≥ (

v

(

k

)

(

y

),

y

−

z

), that is,

v

(

k

)

(

y

) is a subgradient of the concave non-differentiable function

f

(

k

)

(

y

). Vector

v

(

y

) = (

v

(1)

(

y

), …,

v

(

K

)

(

y

)) is a subgradient of function , that is,

F

(

y

) −

F

(

z

) ≥ (

v

(

y

),

y

−

z

).

Therefore, the procedure [1.10] is a specific subgradient algorithm for maximizing the (continuously) non-differentiable concave function F(y).

The following proposition shows that ys converges (non-monotonic convergence) to an optimal linking vector y*, maximizing F(y) subject to joint constraints [1.5]. Let us denote this feasible set by Y. Assume that

The feasible set

Y

is bounded.

Step size

ρ

s

satisfies the conditions:

ρ

s

≥ 0, , , say

p

s

= 1/

s

.

Then lim ys ∈ Y* for s → ∞.

The following sequence of ρs satisfies the conditions of the theorem: ρs = γs/s, for some positive constants and .

Joint constraints [1.5] may have the following form:

[1.11]

with some matrices M(k). However, problems [1.1]–[1.4] and [1.11] can be reformulated similar to problems [1.1–1.5]. Let us define vectors y(K+k) such that M(k)x(k) = y(K+k), k = 1, …, K. Now it is possible to rewrite [1.11] as and after some renotation, derive the problem in the form [1.1–1.5].

1.2.5. Distributed computation

The linkage algorithm can be summarized as follows. Imagine there is a network of distributed computers connecting submodels, say sectors, with a computer of a social planner. At the initial step, sectors k, k = 1, …, K, use arbitrary chosen vectors y0(k) of resource quotas. They submit the information on y0(k) to the central computer. The computer updates quotas y0 = (y0(1), …, y0(K)) by projecting them onto the set Y defining a first feasible approximation y1 = (y1(1), …, y1(K)). All sectors independently solve models [1.1–1.4] with resource quotas y1, calculate shadow prices v1(k) of common resources (constraint [1.4]) and submit to the central computer. The central computer calculates y1 + ρ1v1 with a step-size ρ1 such that the product ρ1v1 corresponds to the scale of y1. Vector y1 + ρ1v1 is projected onto Y to derive quotas y2. At the iteration s + 1, the algorithm derives the next approximation of quotas ys+1 = (ys+1(1), …, ys+1(K)) by shifting ys in the direction of vector vs = (vs(1), …, vs(k)), according to procedure [1.10].

At each iteration, all sectors independently calculate stopping criteria εk(s) = (b(k), us(k) (ys)) + (ys(k), vs(k)(ys)) − wk(c(k), xs(k)(ys)) and submit values εk(s) to the central computer. If , where ε is an admissible accuracy, then the algorithm stops. Otherwise, it continues further. The convergence of the parallel independent optimization (linkage) of sectoral/regional models according to this algorithm, without revealing sectoral/reginal information, is possible due to the additional requirement . This allows us to prove the convergence of solutions (linkages) ys rather than the convergence of objective function F(ys). The convergence of the proposed linkage algorithm under ASI is based on the theory of (continuously) non-differentiable optimization (for more details, please see Ermoliev (1980)).

1.3. Linking energy and agricultural models for FEW nexus

The proposed iterative algorithm has been applied for linking energy and agricultural sectoral models under joint constraints on water and land use. Both models can be used for optimal energy and agricultural production and allocation planning. In the following, we only briefly outline the models. Further details can be found in some previous studies (Havlik et al. 2011; Ermolieva et al. 2016, 2021a; Gao et al. 2018; Ermoliev et al. 2022; Zagorodny et al. 2024). The models are spatially explicit, which allows us to link the models across locations and thus control local drivers having significant implications on the overall results of models’ integration.

The energy model (Ermoliev et al. 2023) incorporates the main stages of energy flows from resources to demands: energy extraction from energy resources, primary energy conversion into secondary energy forms, transport and distribution of energy to the point of end and conversion into products for end users to fulfill specific demands. The structure of the model is such that it can incorporate various energy resources such as coal, gas, crude oil and renewables. Primary energy sources include coal, crude oil, gas, solar and wind; secondary energy sources are fuel oil, methanol, hydrogen, electricity, ammonia, etc.; final energy products are coal, fuel oil, gas, hydrogen, ammonia, methanol, electricity, etc. Demands for useful energy products come from main sectors of the economy: industrial, residential, transport, agricultural, water and energy. Each technology is characterized by unit costs, efficiency, lifetime, emissions, etc. Additional sectoral (and cross-sectoral joint) constraints are imposed to capture the requirements and the limitations on the natural resource use and availability, and investments. The model can include the existing technologies, as well as the new zero-carbon green technologies, at the beginning of implementation or even in the research stage, for example, various renewable and carbon capturing technologies.

The agricultural model (Havlik et al. 2011; Ermolieva et al. 2016, 2021a) can include main crops and livestock production and management systems, characterized by systems-specific production costs, water and fertilizers requirements, emission factors and other parameters. The supply of crops and livestock products need to cover food, feed and biofuel demands and fulfill security constraints. The food security constraint requires that the energy and nutrients consumption from grain and livestock products is not less than the required kilocalories and nutrients needed to satisfy dietary requirements in cereals, vegetable and animal products (meat and dairy products). Livestock feeds fulfill the livestock dietary requirements in energy intake measured in megacalories. Biofuels production from crops (and agricultural residues) have to fulfill biofuel mandates. In the model, land uses comprise agricultural (crop and pasture) land, grass land and natural land. Land use changes can be regulated by setting regulatory constraints on land expansion and conversion. Security constraints introduce competition for limited natural resources (land and water) among different land uses.

Energy and agricultural sectors compete for common land and water resources. Assume that regional planners, decision makers and sectoral authorities pursue a goal to minimize costs and maximize profits from energy and agricultural production under various joint balance (supply–demand) and resource constraints to fulfill the energy and agricultural demands. Namely, the goal is to choose a portfolio of energy technologies to be installed and operated to produce, convert and transfer energy products among locations, and a portfolio of agricultural technologies and management systems to produce and transfer among locations, agricultural commodities fulfilling constraints on natural resources, environmental pollution and end-products demands. The models include relevant risk-related systems performance criteria. These performance measures enable better understanding of how systems (individually and jointly) can perform in an uncertain environment in the presence of climate change, weather variability, market uncertainties, etc. The understanding of interdependencies of energy–water–agricultural systems can motivate regional and sectoral planners, experts, and involved stakeholders to make cross-sectoral coherent and risk-adjusted robust decisions.

1.3.1. Energy, water and agricultural security nexus: case study in China

In the case study in Shanxi province, China (Gao et al. 2018), the energy model was customized to include only coal-based industries and processes, that is, mining, washing, chemical production and power generation. The coal-based technologies consume vast amount of water, for example, for coal mining and washing, coal power plants cooling and steam production.

In the context of coal-based energy production, the integrated energy–agricultural–water model is formulated as follows. The problem is to decide how much coal xijlmt of type i, i = 1, …, I, to extract in location j, j = 1, …, J, transport to location m, m = 1, …, M, and convert by technology t, t = 1, …, T, to cover coal-based products (electricity, heating, cooling, pharmaceutical industry, etc.) demands.

Agricultural model decisions zkjm concern agricultural commodities production, k = 1, …, K, in location j and transported to location m. The overall goal is to minimize the total costs of energy and agricultural production, transportation and conversion.

Individual sectoral goal functions are formulated as follows:

[1.12]

and

[1.13]

for energy [1.12] and agricultural [1.13] sectors. Production costs define all components of coal production costs, including extraction and washing, of a unit (ton) coal of type i in location j, transportation costs represent all costs associated with transporting a unit coal i from location j to location m, define conversion costs of a unit coal i by technology t in location j, denote agricultural production cost per unit (ton) agricultural commodity k in location j and stand for transportation costs of a unit agricultural commodity k from j to m, i = 1, …, I, j = 1, …, J, m = 1, …, M, t = 1, …, T, k = 1, …, K.

The energy security constraints ensure that the demands for coal-based end products (electricity, heat, coke, gas and oil) are fulfilled:

[1.14]

where denotes the conversion coefficient of coal i in location j by technology t, and stands for the end product d demand.

Agricultural production fulfills food security constraints, which can be calculated according to nutrients and kilocalories norms for population by age groups:

[1.15]

where is the demand for agricultural commodity k in location m to meet food security requirements.

Sectoral land use constraints

[1.16]

and

[1.17]

incorporate land demand by coal [1.16] and crop [1.17] production activities, where and are land use constraints for coal and agricultural sectors in location j, respectively. Parameter Sij in [1.16] defines the area that may become unusable as a result of coal mining, Δlj is a portion of agricultural land overlapping with a coal field in location j, parameter rij for coal i in location j defines plausible land reclamation rate and gij is a parameter allowing us to calculate the land under coal reject material. In equation [1.17], parameter lkj defines the area required for a unit crop k production in location j. Equation [1.18] introduces the restriction on the total land use in location j by energy and agricultural sectors

[1.18]

Total available water Wj for both sectors and sectoral water quotas ( and ) significantly affect the choice of coal and crop (energy) production technologies through water utilization constraints:

[1.19]

and

[1.20]

where and are quotas on water use by coal and agricultural activities in location j, defines water requirement for a unit coal i production in location j, is water required for a unit coal i conversion in location j and is water required for a unit crop k production in location j. Water use for coal and for agricultural production are constrained by total water Wj available in j:

[1.21]

In the condition of ASI, the planner does not have full information about separate LP energy ([1.12], [1.14], [1.16] and [1.19]) and agricultural ([1.13], [1.15], [1.17] and [1.20]) submodels. To link the models under joint constraints [1.18] and [1.21], we implement procedure [1.10]. At the initial step of the procedure s = 0, individual sectoral models are solved using initial sectoral land and water quotas , and , . Resource quotas at step s are adjusted according to [1.10] using shadow prices (dual variables) of energy and agricultural sectors land and water resource constraints

[1.22]

[1.23]

[1.24]

[1.25]

and constraints [1.18] and [1.21].

1.3.2. Selected results

Results were calculated and compared for three scenarios: (1) separately optimized energy and agricultural models, (2) hard-linked IM (one-code model) and (3) separate models integrated via the linkage procedure [1.10]. In the first scenarios, the net profits can be higher than in other two scenarios because the sectors are not restricted by joint constraints. This result is misleading the resource allocation analysis. In scenario 3, the solution of the iterative linkage process converges rather quickly (only in 10 iteration steps) to the solutions of the hard-integrated model, scenario 2.

Figure 1.1.Convergence of the iterative linking procedure in terms of the goal function values F(ys). Vertical axis displays net profits, and the iteration step is marked on the horizontal axis. The three curves (Scen1, Scen2 and Scen3) correspond to three different initial land and water quota scenarios at s = 0.

Figure 1.1 illustrates the non-monotonic convergence of the linkage algorithm for three different scenarios of initial y0 quotas allocated to energy and agricultural sectors. The choice of the step-size ρs in [1.10] affects the convergence rate, and the value of the product ρsvs must correspond to the value of solutions ys.

1.4. Conclusion

In this chapter, we discuss the problem of linking distributed individual sectoral and/or regional optimization models into an inter-sectoral IM. The approach for linking models is based on an iterative SQG procedure of, in general, non-smooth non-differentiable optimization converging to a socially optimal solution maximizing an implicit nested non-differentiable social welfare function. The convergence of the algorithm relies on the duality theory and the non-differentiable optimization. The iterative solution procedure can be used for robust estimation and machine learning problems, in particular, it can be viewed as an endogenous reinforced learning problem.

The iterative SQG-based methods and their stochastic versions are intended for robust optimization of deterministic and stochastic systems with large number of decision variables and scenarios of uncertainties due to the ability of these methods to link scenario-simulation and optimization procedures. The proposed linkage method will be further developed for linking stochastic models enabling integrated management of global systemic risks.

The linkage of models is, in a sense, opposite to decomposition methods. While in the decomposition (e.g. Dantzig and Wolfe 1960; Kim and Nazareth 1991), we split an existing integrated optimization model into a number of smaller sub-models, in the linkage, we obtain an IM of the system by linking existing explicitly unknown sub-models. The proposed linkage procedure provides a flexibility enabling the simultaneous use of linkage and decomposition procedures; in other words, endogenously disaggregating models to make their further integration (linkage) more efficient.

1.5. Acknowledgments