Kuhn Tucker Optimality Conditions Research Articles

SOME CHARACTERIZATION RESULTS FOR NONSMOOTH INTUITIONISTIC FUZZY OPTIMIZATION VIA SYMMETRIC gH-DIFFERENTIABILITY

In recent years, there has been a growing interest in optimization problems with uncertainty. Intuitionistic fuzzy optimization is one of the approaches for investigating such real-world optimization problems that contain uncertain data and, therefore, that are not well defined. Nonetheless, there is not enough discussion on the optimality conditions for optimal solutions in intuitionistic fuzzy optimization problems (IFOPs) and methods for solving such uncertain extremum problems. Although the derivative notion is a very important property of optimization models, there are many real-world problems and processes with uncertainty that cannot be modeled as differentiable intuitionistic fuzzy functions (IFFs). In this work, therefore, a new notion of generalized derivative is defined. Namely, the definition of symmetric gH-derivative is introduced for nonsmooth IFOPs and some properties of this concept are analyzed. Further, the Karush–Kuhn–Tucker optimality conditions are established for an optimal solution in an intuitionistic fuzzy extremum problem with symmetric gH-differentiable IFFs and inequality constraints. Meanwhile, several examples are given to illuminate the obtained results.

Coordinating a bi‐level blood supply chain with interactions between supply‐side and demand‐side operational decisions

AbstractIn most blood supply chains, blood centers and hospitals make individual decisions, resulting in an inefficient structure of the blood supply chain, which in turn renders supply and demand matching a challenging exercise. In this work, we make the very first attempt to optimize the interaction between blood centers and hospitals. To that end, this paper investigates collection, production, replenishment, issuing, inventory, and wastage decisions under three different blood supply chain channel structures, that is, the decentralized, centralized, and coordinated structures. We propose a bi‐level optimization program to model the decentralized system and use the Karush–Kuhn–Tucker optimality conditions to solve that. In such a system, hospitals tend to order more than their actual need, resulting in overcollection, overproduction, and high wastage rates. On the other hand, in a centralized system decisions are made by a central decision‐maker, which results in higher performance. Recognizing the challenges of implementing a centralized system, we design a novel coordination mechanism to motivate hospitals to operate in a centralized system. Analysis of a case study in Canada indicates that integration can significantly improve the performance of system; allowing substitution between blood products can decrease the total cost of the blood supply chain by 14.41%; an increase in supply or decrease in demand can be detrimental under inappropriate structure, facilitating coordination mechanism; offering subsidy beyond a threshold is not beneficial to the blood centers.

Optimal product pricing and lot sizing decisions for an omni-channel supply chain

In an omni-channel supply chain, companies offer customers multiple channels to interact with their brand. This allows for greater outreach and sales, resulting in increased customer loyalty and trust. Furthermore, there has been little discussion on the simultaneous consideration of strategic and tactical decisions under return policies in omni-channel supply chains. To maximize total profit, this research proposes a two-echelon supply chain, comprising a manufacturer and multiple retailers, including traditional, electronic and omni-channels. The non-deterministic demand function takes into account the sales price, refund price, delivery time and order quantities. A heuristic algorithm based on analysis of Karush–Kuhn–Tucker optimality conditions used to solve a nonlinear programming model. This research presents an analysis of the factors influencing the decision variables and profit of the entire supply chain and offers retailers a unique way to evaluate their performance and make operational decisions regarding price, delivery time and order quantities.

Capacity market in the decarbonization era: Adequacy, flexibility, and environmental revenue

AbstractWith the increasing penetration rate of renewable energy in the power system, it becomes increasingly important to coordinate adequacy, flexibility, and environmental protection. The development of the power market provides diverse signals for power investors and plays an increasingly significant role in the decarbonization of the power system. However, due to the interdependent operations of various power markets, measuring the multifaceted value of power generation within traditional market mechanisms poses considerable challenges. Herein, a bi‐level optimization model is proposed to integrate the capacity remuneration mechanism (CRM), energy market (EM), and carbon emission trading (CET). The adequacy, flexibility, and environmental revenue expectations of different generation resources through synergistic transactions are derived using Lagrange multipliers. Furthermore, the bilevel optimization model is converted to its equivalent single‐level scheme based on the Karush–Kuhn–Tucker optimality conditions. Finally, several case studies are conducted to demonstrate the effectiveness of the proposed model, along with comparisons and sensitivity analyses to study the principles behind synergistic transactions and the impact of some key parameters.

An expansion planning model for strategic visioning of active distribution network in the presence of local electricity market

AbstractEmergence of different stakeholders with contrary objectives in active distribution network (ADN) requires for a fair network expansion planning that considers all of the contrary objectives. This paper provides an expansion planning model for ADN strategic visioning from electric distribution stakeholders' organization (EDSO) perspective, which takes into account the objectives of all stakeholders. The stakeholders are network owner, active customers (ACs), consumers, and environmental regulatory body (ERB). The proposed model is multistage bi‐level, where the upper‐level minimizes the present value of the total costs imposed to all ADN stakeholders and the lower‐level minimizes the total generation costs within the distribution‐level electricity market. The Karush–Kuhn–Tucker optimality conditions and the linearization techniques are employed through which the model is converted to mixed integer linear programming which can be tackled effectively. The model provides a visionary expansion plan that policymaker must legislate to realize it. Effectiveness of the model in providing a visionary expansion plan considering EDSO's perspective in making cost effective planning decisions per social welfare goals is verified through several case studies on 33‐bus standard test system. According to the simulation results, transition from DSO to EDSO in the presence of low‐capacity ACs improves social welfare by 5%. But this improvement is up to 70% in the presence of high‐capacity ACs.

Co-optimization of multi-stakeholders in multi-heterogeneous energy system considering gas flow dynamics

Multi-heterogeneous energy system aims to exploit the advantages of coupling between electricity, heat, and gas. However, the common approach utilized to obtain the overall optimum cannot consider the existence and interests of all stakeholders in energy sectors such as electricity, heat, gas, and energy hubs. In contrast, this paper develops collaborative optimization and solution methods after clarifying the interaction mechanism of multi-stakeholders. Firstly, the operational models of natural gas system, electric power system, heating pipeline system, and energy hub are built with emphasis on the implementation of dynamic modeling of the gas network. Secondly, the multi-stakeholder interface constraints are established, while an upper-mid-lower optimization mechanism is proposed. Then, to solve the structured dynamic model, a microelement-integral approach for solving dynamic gas flows is proposed and extended to entire gas network. Moreover, a set of methods including Karush–Kuhn–Tucker optimality conditions, the big-M method, the quasi-accurate method, and the stepped-price extension method are employed to solve multi-stakeholder optimization problems with multiple nonlinear terms. Finally, the simulation case with six comparison scenarios are constructed to validate the proposed model and methods. The obtained results show that the gas network dynamic model can reflect the actual transmission mechanism of gas flow, and the proposed multi-stakeholder model can effectively balance the interests of various stakeholders. Consequently, all stakeholders have more potential to jointly promote the application and construction of multi-heterogeneous energy systems.

Semi-infinite interval equilibrium problems: optimality conditions and existence results

This paper aims to obtain new Karush–Kuhn–Tucker optimality conditions for solutions to semi-infinite interval equilibrium problems with interval-valued objective functions. The Karush–Kuhn–Tucker conditions for the semi-infinite interval programming problem are particular cases of those found in this paper for constrained equilibrium problem. We illustrate this with some examples. In addition, we obtain solutions to the interval equilibrium problem in the unconstrained case. The results presented in this paper extend the corresponding results in the literature.

Solving Bilevel Optimal Bidding Problems Using Deep Convolutional Neural Networks

Current state-of-the-art solution techniques for solving bilevel optimization problems either assume strong problem regularity criteria or are computationally intractable. In this article, we address power system problems of bilevel structure, commonly arising after the deregulation of the power industry. Such problems are predominantly solved by converting the lower level problem into a set of equivalent constraints using the Karush–Kuhn–Tucker optimality conditions at an expense of binary variables. Furthermore, in case the lower level problem is nonconvex, the strong duality does not hold rendering the single-level reduction techniques inapplicable. To overcome this, we propose an effective numerical scheme based on bypassing the lower level completely using an approximation function that replicates the relevant lower level effect on the upper level. The approximation function is constructed by training a deep convolutional neural network. The numerical procedure is run iteratively to enhance the accuracy. As a case study, the proposed method is applied to a price-maker energy storage optimal bidding problem that considers an ac power flow-based market clearing in the lower level. The results indicate that greater actual profits are achieved as compared to the less accurate dc market representation.

A proactive strategy for virtual power plants including multiple private owners equipped with energy storages

This paper presents a novel framework for the optimal operation strategy of a virtual power plant (VPP) comprises a set of various stakeholders such as renewable energy sources, distributed energy resources with private owners, and flexible loads in day-ahead and real-time energy markets. In order to model the strategic behavior of private owners within the VPP geographic location, a proactive stochastic bi-level model is developed, where the upper level assigns to minimize expected operation cost of the technical VPP and the lower level associates with maximizing the private owner's profits. Note that, the uncertainty set encompasses the real-time energy market price as well as renewables power production. Afterward, the proposed bi-level model is converted to a mixed-integer linear programming problem through Karush–Kuhn–Tucker optimality conditions along with strong duality theory. Moreover, the impacts of implementation of three different pricing schemes including fix pricing, time of use, and real-time pricing on the market trading strategy of VPP and also its interactions with private owners is investigated through comparative case studies. The obtained results demonstrate that the proposed approach could allow both VPP aggregator and private owners to take the advantages of the market conditions.

Perturbed‐analytic direct transcription for optimal control (PADOC)

AbstractThe herein coined PADOC (perturbed‐analytic direct transcription for optimal control) stands for a new transcription method for direct trajectory optimization. We construct PADOC on a novel segmented decomposition method that provides series solutions for nonlinear problems. To transcribe the infinite dimensional problem into a finite one, PADOC comes along with a new solution approach, namely, the herein coined average nonlinear programming (aNLP). The aNLP theorem suggests generating a staircase optimal solution (low resolution) and turning it into a distributed optimal solution (high resolution) by exploiting the Hamiltonian of the problem. The analytic architecture provides PADOC with an analytic connection of a truncated order between the discrete nodes of the solution. This renders stability, accuracy within an analytic resolution, robustness, and a cut‐down in the number of the decision variables in the frame of NLP. We also prove that the multipliers associated with the Karush–Kuhn–Tucker optimality conditions in the frame of NLP (as transcribed by PADOC) correspond to a backward analytic solution for the costate equations. We finally show distinct features of PADOC through some examples.

Optimal Implementation Parameters of a Nonlinear Electrical Impedance Tomography Method Using the Complete Electrode Model

This study discusses a nonlinear electrical impedance tomography (EIT) technique under different analysis conditions to propose its optimal implementation parameters. The forward problem for calculating electric potential is defined by the complete electrode model. The inverse problem for reconstructing the target electrical conductivity profile is presented based on a partial-differential-equation-constrained optimization approach. The electrical conductivity profile is iteratively updated by solving the Karush–Kuhn–Tucker optimality conditions and using the conjugate gradient method with an inexact line search. Various analysis conditions such as regularization scheme, number of electrodes, current input patterns, and electrode arrangement were set differently, and the corresponding results were compared. It was found from this study that the proposed EIT method yielded appropriate inversion results with various parameter settings, and the optimal implementation parameters of the EIT method are presented. This study is expected to expand the utility and applicability of EIT for the non-destructive evaluation of structures.

A combined Control by Interconnection—Model Predictive Control design for constrained Port-Hamiltonian systems

This paper proposes a Control by Interconnection design, for a class of constrained Port-Hamiltonian systems, which is based on an associated Model Predictive Control optimization problem. This associated optimization problem allows to consider both state and input constraints simultaneously. Based on the first order Karush–Kuhn–Tucker optimality condition, the primal–dual gradient method is then used to build a passive feedback controller, derived from the MPC-induced optimization problem. The resulting passive controller is coupled with the original Port-Hamiltonian system through a power-preserving interconnection, in order to guarantee both the closed-loop stability and constraints satisfaction, but not the optimality anymore. Comments on parameters tuning for the proposed control design, together with validations of the approach through simulations first on a linear LC circuit, then on a nonlinear Permanent Magnet Synchronous Motor and comparisons with a classical MPC design, are provided to discuss the effectiveness of the approach. • A controller combining CbI and MPC is proposed to tackle PHs with constraints. • The MPC principle is used to deal with both state and input constraints. • The CbI takes advantage of the PHs passivity for convergence and stability analysis. • The primal–dual gradient method significantly reduces the computational effort. • The explicit form of the controller facilitates implementation and maintenance.

Expansion planning of a price-maker virtual power plant in energy and reserve markets

This paper addresses the expansion planning problem of a virtual power plant considering the possibility of building new assets such as conventional, renewable, and energy storage units. The VPP is modeled as a price-maker player that strategically participates in energy and reserve markets by altering the prices of these markets to its own benefit. The VPP includes flexible demands, conventional and renewable generating units, and energy storage units. Variabilities in renewable production levels are modeled using a modified clustering K-medoids method, while the uncertain reserve request coefficients are represented using a set of scenarios. The present problem is formulated as a two-stage stochastic bi-level model, where the upper-level problem represents the expected profit of the VPP to be maximized by building new assets, while the lower-level problems model the clearing of the energy and reserve electricity markets for different market conditions, which aim to maximize the social welfare and reduce the cost of the ancillary systems, respectively, and determine the market prices of both markets. The stochastic bi-level model is converted to a single-level mixed-integer linear programming problem by replacing the lower-level problems by their Karush–Kuhn–Tucker optimality conditions and by applying exact linearization techniques. Numerical results show that considering the VPP as a price-maker player influences its expansion decisions and allows the VPP to increase its profit up to 108.8% in comparison with a price-taker behavior. Also, the numerical results indicate that the participation in the reserve market increases a 26.76% the expected profit achieved by the VPP.

A Convex Neural Network Solver for DCOPF With Generalization Guarantees

The dc optimal power flow (DCOPF) problem is a fundamental problem in power systems operations and planning. With high penetration of uncertain renewable resources in power systems, DCOPF needs to be solved repeatedly for a large amount of scenarios, which can be computationally challenging. As an alternative to iterative solvers, neural networks are often trained and used to solve DCOPF. These approaches can offer orders of magnitude reduction in computational time, but they cannot guarantee generalization, and small training error does not imply small testing errors. In this work, we propose a novel algorithm for solving DCOPF that guarantees the generalization performance. First, by utilizing the convexity of the DCOPF problem, we train an input convex neural network. Second, we construct the training loss based on Karush–Kuhn–Tucker optimality conditions. By combining these two techniques, the trained model has provable generalization properties, where small training error implies small testing errors. In experiments, our algorithm significantly outperforms other machine learning methods.

Using ρ-cone arcwise connectedness on parametric set-valued optimization problems

Within the inquiry about work, we explore a parametric set-valued optimization problem, where the objective as well as constraint maps are set-valued. A generalization of cone arcwise associated set-valued maps is presented named ρ-cone arcwise connectedness of set-valued maps. We set up adequate Karush–Kuhn–Tucker optimality conditions for the problem beneath contingent epiderivative and ρ-cone arcwise connectedness presumptions. Assist, Mond–Weir, Wolfe, and blended sorts duality models are examined. We demonstrate the related theorems between the primal and the comparing dual problems beneath the presumption.

Constraint qualifications in terms of convexificators for nonsmooth programming problems with mixed constraints

The main aim of the paper is to introduce certain constraint qualifications for a nonsmooth programming problem in terms of semi-regular convexificators and investigate their relations with other existing notions of constraint qualifications. The programming problem under consideration has mixed constraints, that is, it involves both inequality and equality constraints. All these notions are in terms of upper semi-regular convexificators of inequality constraints and pseudo-differentials of equality constraints. Based on a sufficient condition for error bound property, the implication relation between quasinormality and error bound property in terms of convexificators is investigated in this paper. Three conditions are introduced, namely constant positive linear dependence condition (CPLD), constant rank constraint qualification (CRCQ) and Mangasarian–Fromovitz constraint qualification (MFCQ) in terms of convexificators. These conditions are in fact shown to be constraint qualifications as Karush–Kuhn–Tucker optimality conditions hold when CPLD holds and both MFCQ and CRCQ imply CPLD. Further, it is observed that CPLD and quasinormality conditions are independent for nonsmooth problems in terms of convexificators.

Slater Condition for Tangent Derivatives

Noting that the existing Slater condition, as a fundamental constraint qualification in optimization, is only applicable in the convex setting, we introduce and study the Slater condition for the Bouligand and Clarke tangent derivatives of a general vector-valued function F with respect to a closed convex cone K. Without any assumption, it is proved that the Slater condition for the Clarke (respectively, Bouligand) tangent derivative with respect to K is always stable when the objective function F undergoes small Lipschitz (calm) perturbations. Based on this, we prove that if the Clarke (Bouligand) tangent derivative of F satisfies the Slater condition (with respect to K) then the conic inequality determined by F has a stable metric subregularity when F undergoes small Lipschitz (calm regular) perturbations. In the composite-convexity case, the converse implication is also proved to be true. Moreover, under the Slater condition for the tangent derivative of F, it is proved that the normal cone to the sublevel set of F can be formulated by the subdifferential of F, which improves the corresponding results in either the smooth or convex case. As applications, without any qualification assumption, we improve and generalize formulas for the normal cone to a convex sublevel set by Cabot and Thibault [(2014), Sequential formulae for the normal cone to sublevel sets. Transactions of the American Mathematical Society 366(12):6591–6628]. With the help of these formulas, some new Karush–Kuhn–Tucker optimality conditions are established.

Transaction Model Based on Stackelberg Game Method for Balancing Supply and Demand Sides of Multi-Energy Microgrid

To improve the coordination and complementarity of multiple energy sources, balancing the interests of different participants in a multi-energy system is of great importance. However, traditional centralized optimization can hardly reflect the game relationship between supply side and demand sides. A trading model based on the Stackelberg game model is proposed in this paper to balance the interests of the supply side and demand side and reduce the carbon emissions. First of all, the process of trading between the supply side and demand side based on smart contracts is described. A contractual consensus is obtained through an internal game, and the transaction is completed automatically. Secondly, a bilevel optimization model is established to coordinate the benefits of both parties based on the Stackelberg game model. The energy operator acts as a leader, and considers the two objectives, i.e., maximizing net income and minimizing carbon emissions, and uses the linear weighting method to convert the dual objectives into single objective. Users act as followers and aim to increase the comprehensive benefits, including energy cost and comfort. Then, Karush–Kuhn–Tucker optimality condition is used to transform the bilevel optimization model into an equivalent single-level model. Finally, simulation results show that the proposed method can coordinate the economic interests of both sides of supply and demand and effectively reduce the carbon emissions of the energy operator.

Multiobjective approximate gradient projection method for constrained vector optimization: Sequential optimality conditions without constraint qualifications

In this paper, we establish multiobjective approximate gradient projection (MAGP) and linear multiobjective approximate gradient projection (LMAGP) sequential optimality conditions without constraint qualification for multiobjective constrained optimization problems. Further, we introduce constraint qualifications under which a point that satisfies established optimality conditions also satisfies Karush–Kuhn–Tucker optimality conditions. Such constraint qualifications are called strict constraint qualifications. We discuss the relationship between introduced constraint qualifications and validate it by suitable examples.

An Exact Algorithm Based on the Kuhn–Tucker Conditions for Solving Linear Generalized Semi‐Infinite Programming Problems

Optimization problems containing a finite number of variables and an infinite number of constraints are called semi‐infinite programming problems. Under certain conditions, a class of these problems can be represented as bi‐level programming problems. Bi‐level problems are a particular class of optimization problems, in which there is another optimization problem embedded in one of the constraints. We reformulate a semi‐infinite problem into a bi‐level problem and then into a single‐level nonlinear one by using the Kuhn–Tucker optimality conditions. The resulting reformulation allows us to employ a branch and bound scheme to optimally solve the problem. Computational experimentation over well‐known instances shows the effectiveness of the developed method concluding that it is able to effectively solve linear semi‐infinite programming problems. Additionally, some linear bi‐level problems from literature are used to validate the robustness of the proposed algorithm.