Duality Theorem Research Articles

Load Shifting Demand Response in Energy Scheduling Based on Payment Cost Minimization Auction Mechanism

Demand response (DR) is proven to be very efficacious for load mitigation especially in peak time period. On the other hand, DR facilitates both consumers, system operator as well as producers to moderate their payments while reducing system operating costs. Offer cost minimization is currently used as the clearing mechanism associated with locational marginal pricing scheme to determine the consumers’ payments. Such clearing and pricing mechanisms are inconsistent as the system costs is being minimized, while the payments are calculated based upon marginal prices. Payment cost minimization auction as a price-based clearing mechanism is envisaged to be an effective alternative to solve such a crucial issue. This paper shows how to include DR in PCM mechanism to further reduce the consumers’ payment. It facilitates utilizing price responsive consumers for load shifting DR in a PCM auction. The optimization problem is modeled as a mixed-integer nonlinear bi-level programming. Duality theorem, KKT conditions and integer algebra are used to convert such a complicated optimization problem to a single level MILP problem. This problem is then solved by CPLEX in GAMS. The impacts are studied by implementing the proposed formulation to solve the clearing problem in the case studies deriving promising numerical results.

Умови існування екстремального елемента для задачі відшукання відстані між двома множинами, єдиності екстремального елемента еквівалентної їй задачі, властивості функції відстані

The theory of approximation of a function is a direction of mathematics which is intensively developing. The work of P. L. Chebyshev in 1857 is considered the beginning of the modern theory of approximations. It is devoted to polynomials that deviate the least from zero. In this work, the concept of the best approximation was introduced. Later, problems were investigated in which individual functions approached with polynomials, trigonometric polynomials, rational functions, etc. in different metrics. These tasks are a partial case of the problem of the best approximation of an element of linear normed space by convex set of this space. General theorems of existence, uniqueness of an extremal element, properties of the best approximation functional, duality theorems and criteria of an extremal element for this problem are established [1]. The more general problem are problem of finding the distance between two sets of linear normalized space is also considered [2, 3]. In [4, 5] the relations of duality, criteria of extremal element and sequence are proved for this problem. In this article established the conditions of the existence of an extremal element for the problem of finding the distance between two sets of linearly normalized space, the conditions of the unity of an extremal element for its equivalent problem, the properties of the function of the distance and formulas for finding an extremal element for the problem of finding the distance between two closed spheres of this space.

Utility Maximization with Proportional Transaction Costs Under Model Uncertainty

We consider a discrete time financial market with proportional transaction costs under model uncertainty and study a numéraire-based semistatic utility maximization problem with an exponential utility preference. The randomization techniques recently developed in Bouchard, Deng, and Tan [Bouchard B, Deng S, Tan X (2019) Super-replication with proportional transaction cost under model uncertainty. Math. Finance 29(3):837–860.], allow us to transform the original problem into a frictionless counterpart on an enlarged space. By suggesting a different dynamic programming argument than in Bartl [Bartl D (2019) Exponential utility maximization under model uncertainty for unbounded endowments. Ann. Appl. Probab. 29(1):577–612.], we are able to prove the existence of the optimal strategy and the convex duality theorem in our context with transaction costs. In the frictionless framework, this alternative dynamic programming argument also allows us to generalize the main results in Bartl [Bartl D (2019) Exponential utility maximization under model uncertainty for unbounded endowments. Ann. Appl. Probab. 29(1):577–612.] to a weaker market condition. Moreover, as an application of the duality representation, some basic features of utility indifference prices are investigated in our robust setting with transaction costs.

Operational scheduling of intercity passenger transportation company participating in energy and reserve markets

The electric vehicle (EV) is becoming the main mode of intercity passenger transportation. The intercity passenger transportation company (IPTC) is envisioned to participate in the electricity market to maximize the operation profit. Hence, it is necessary to study some strategic operating behaviors of the IPTC, including EV scheduling and bidding, to ensure the efficient operation of intercity passenger transport. This paper develops a decision-making tool based on a bi-level programming model for a merchant price-maker IPTC to determine the most beneficial scheduling and trading actions in the day-ahead market. The upper level of the model considers the mobility characteristics of EVs by a modified time–space network (TSN) to help the IPTC formulate intercity passenger shifts and bidding strategies. In this level, the IPTC maximizes its profit in the day-ahead market through bidding strategies, transportation scheduling, and charging/discharging arrangement at each station. In the lower level of the model, the joint energy and reserve market is cleared with maximum social welfare. The bi-level model can be transformed into an easily solved mixed-integer linear programming problem by applying the strong duality theorem and a binary expansion method to handle the nonlinear constraints of the reformulated single-level problem based on Karush-Kuhn-Tucker theorems. Finally, case studies are carried out on a modified IEEE 39-bus system and a practical 714-bus power system in China. The numerical results indicate that the IPTC tends to schedule more EVs to park in cities directly connected to congested transit lines and increase the number of discharging vehicles, resulting in the growth of the scheduled energy of discharging in these cities and thus higher profits. The different operation strategies of the IPTC are also discussed in the day-ahead market.

Dynamic transactive energy in multi-microgrid systems considering independence performance index: A multi-objective optimization framework

This paper proposes a leader multi-follower optimization for energy transactions in the multi-microgrid systems. The transactions of microgrids with the distribution network operator can be provided by energy internet. The distribution network operator can buy/sell energy from/to the microgrids. The upper-level of the optimization is modeled as a multi-objective optimization in which the profit of distribution network operator, independence performance, and energy not supplied of the system have been optimized using the lexicography approach. In the lexicography approach, two-steps for objectives are defined. In the first step, the profit of the distribution network operator is maximized, while the independence performance index and energy not supplied of system have been optimized in the second step. In the lower-level, microgrids are considered as followers to minimize their costs. The uncertainty of renewable generation and market prices are applied to the model using scenario-generation and scenario-reduction technique. The proposed non-linear bi-level model is transformed to the linear single-level optimization using the Karush-Kuhn-Tucker optimality conditions and strong duality theorem. The proposed model is tested on the standard case study and the results show the efficiency of the proposed model in terms of independence index, energy not supplied, and greenhouse gas emissions.

Banach Compactness and Banach Nuclear Operators

In this paper, we introduce the notion of (uniformly weakly) Banach-compact sets, (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators which generalize p-compact sets, p-compact operators and p-nuclear operators, respectively. Fundamental properties are investigated. Factorizations and duality theorems are given. Injective and surjective hulls are used to show the spaces of (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators are quasi Banach operator ideals.

Generalized Farkas Lemma with Adjustable Variables and Two-Stage Robust Linear Programs

In this paper, we establish strong duality between affinely adjustable two-stage robust linear programs and their dual semidefinite programs under a general uncertainty set, that covers most of the commonly used uncertainty sets of robust optimization. This is achieved by first deriving a new version of Farkas’ lemma for a parametric linear inequality system with affinely adjustable variables. Our strong duality theorem not only shows that the primal and dual program values are equal, but also allows one to find the value of a two-stage robust linear program by solving a semidefinite linear program. In the case of an ellipsoidal uncertainty set, it yields a corresponding strong duality result with a second-order cone program as its dual. To illustrate the efficacy of our results, we show how optimal storage cost of an adjustable two-stage lot-sizing problem under a ball uncertainty set can be found by solving its dual semidefinite program, using a commonly available software.

A longitudinal study of agriculture households in Indonesia: The effect of land and labor mobility on welfare and poverty dynamics

The conventional wisdom of Arthur Lewis’s dual sector model says that households in the agricultural (traditional) sector who can move out to a non-agricultural (modern) sector will become better off. We then scrutinize the last three waves of the Indonesia Family Life Survey (IFLS) to analyze the existence of the dual theorem. Our study uses Difference in Difference (DiD) regressions and ordered logit regressions to confirm that moving out of agriculture sectors has significantly increased the welfare of poor agricultural households, especially in the period of 2000–2007, but this is not the case of 2007–2014. Movement out of agricultural sectors decreases the probability of being always poor by 13.5 percentage points. However, when the economy transforms into a more advanced economy, simply moving out of agriculture does not guarantee that farmers, especially landless farmers, will become better off. Welfare improvement requires a shifting to formal non-agricultural sectors, but unfortunately farmers might not be readily equipped with the skills required in formal sectors. Our study also obviously confirms that farmland is an important asset for agricultural households. Agricultural households experiencing a decrease of agricultural land also decreased their expenditure per capita by IDR 36,833 in 2000 and IDR 68,683 in 2007. These findings suggest that, currently, moving out of agriculture is not the solution to improve the well-being of farmers. Keeping farmland ownership, raising investment in human capital, and the modernization of agriculture should be the main concerns in agricultural development.

A Classification of Functions in Multiterminal Distributed Computing

In the distributed function computation problem, dichotomy theorems, initiated by Han-Kobayashi, seek to classify functions by whether the rate regions for function computation improve on the Slepian-Wolf regions or not. In this paper, we develop a general approach to derive converse bounds on the distributed function computation problem. By using this approach, we recover the sufficiency part, i.e. the conditions such that the Slepian-Wolf regions become optimal, of the known dichotomy theorems in the two-terminal distributed computing. Furthermore, we derive an improved sufficient condition on the dichotomy theorem in the multiterminal distributed computing for the class of i.i.d. sources with the positivity condition. Finally, we derive the matching sufficient and necessary condition on the dichotomy theorem in the multiterminal distributed computing for the class of smooth sources.

A Universal and Structured Way to Derive Dual Optimization Problem Formulations

The dual problem of a convex optimization problem can be obtained in a relatively simple and structural way by using a well-known result in convex analysis, namely Fenchel’s duality theorem. This alternative way of forming a strong dual problem is the subject of this paper. We recall some standard results from convex analysis and then discuss how the dual problem can be written in terms of the conjugates of the objective function and the constraint functions. This is a didactically valuable method to explicitly write the dual problem. We demonstrate the method by deriving dual problems for several classical problems and also for a practical model for radiotherapy treatment planning, for which deriving the dual problem using other methods is a more tedious task. Additional material is presented in the appendices, including useful tables for finding conjugate functions of many functions.

Bi-Level Programming for Optimal Operation of an Active Distribution Network With Multiple Virtual Power Plants

Virtual power plants (VPPs) have become an effective technique to manage a growing number of flexible resources, which have posed great technical challenges to distribution system operators (DSOs). This article proposes a bi-level programming approach for the collaborative management of an active distribution network (ADN) with multiple VPPs by designing comprehensive prices for active and reactive power. The upper layer aims to minimize the overall operation cost of the ADN considering the system security and economic operation and the interactions among the power market, ADN and VPPs. The lower layer aims to maximize the benefits of each VPP agent considering various flexible resources. Then, the bi-level model is transformed into a tractable single-level problem by using a linearization method, the Karush–Kuhn–Tucker (KKT) optimality conditions, the Fortuny-Amat transformation and the strong duality theorem. Case analyses indicate that the proposed strategy can effectively enhance the system security and improve the system economic performance by introducing reactive power pricing. The implementation effect and superiority of the proposed strategy are profoundly analyzed under different scenarios and conditions, which indicates its promising application value in the industrial field.

Game-Theory Based V2G Coordination Strategy for Providing Ramping Flexibility in Power Systems

Large-scale integration of renewable generation into power systems invariably affects the system ramping capability. However, the vehicle-to-grid (V2G) concept that allows for using electric vehicles (EVs) as energy storages with the capability of bidirectional energy transfer between the EVs and the grid, can be employed to mitigate the above adverse effect. This paper proposes a game-theory-based V2G coordination strategy that uses EV clusters to improve ramping flexibility in power systems. In the proposed strategy, the V2G concept, representing the interactions between the distribution system operator (DSO) and EV clusters, is formulated as a Stackelberg game. The DSO acts as a leader who decides the charging prices for the buses to which the EV clusters are connected, while the EV clusters simply serve as followers, scheduling their own charging and discharging. This bi-level model is further reduced to a single-level, mixed-integer second-order cone programming (MISOCP) problem based on the Karush-Kuhn-Tucker (KKT) conditions, the strong duality theorem and second-order cone (SOC) relaxation. The performance of the proposed V2G coordination strategy on a modified IEEE 33-bus system connecting EV clusters and PV generations is investigated through simulations, and the results demonstrate that the largest ramp of the system can be reduced by up to 39% when EV clusters are providing flexibility, while the EV clusters can also have greatly reduced charging costs.

Obstructions to weak approximation for reductive groups over p-adic function fields

We establish arithmetic duality theorems for short complexes of tori associated to reductive groups over p-adic function fields. Using arithmetic dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split ones) and relate this obstruction to an unramified Galois cohomology group.

Brauer groups and Néron class groups

Let [Formula: see text] be a global field and let [Formula: see text] be a finite set of primes of [Formula: see text] containing the Archimedean primes. We generalize the duality theorem for the Néron [Formula: see text]-class group of an abelian variety [Formula: see text] over [Formula: see text] established previously by removing the requirement that the Tate–Shafarevich group of [Formula: see text] be finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve [Formula: see text] over [Formula: see text] to a certain finite subquotient of the Brauer group of [Formula: see text].

Bi-Level Optimization Framework for Buildings to Heating Grid Integration in Integrated Community Energy Systems

This paper proposes a bi-level optimization framework for buildings to heating grid (B2HG) integration in integrated heating/electricity community energy systems. The proposed B2HG framework enables buildings to carry out heating demand response to the integrated community energy systems (ICES). In upper level, ICES operator with energy conversion equipment devotes to maximizing its profits by optimizing the energy generation/supply schedules and the sale heating price. In lower level, a detailed physical model of the building with adjustable indoor radiators is developed while considering the building thermal dynamics. Consumers minimize their heating costs by adjusting the flow rates of their radiators according to the heating price provided by the upper level. Then, the proposed bi-level optimization problem is converted into a mixed-integer linear programming using Karush-Kuhn-Tucker conditions and strong duality theorem. The piecewise linearization is used to eliminate the nonlinearity of heating network constraints. Numerical results show that consumers can provide heating demand response to ICES operator by considering the thermal dynamics. With the multiple energy conversion equipment, the ICES operator can unlock the flexibility from the supply side. Moreover, the proposed bi-level optimization method can combine the flexibility from both the supply and demand side, and benefits both the ICES operator and building consumers.

A stochastic dual dynamic programming method for two-stage distributionally robust optimization problems

This paper studies a class of two-stage distributionally robust optimization (TDRO) problems which comes from many practical application fields. In order to set up some implementable solution method, we first transfer the TDRO problem to its equivalent robust counterpart (RC) by the duality theorem of optimization. The RC reformulation of TDRO is a semi-infinite stochastic programming. Then we construct a conditional value-at-risk-based sample average approximation model for the RC problem. Furthermore, we analyse the error bound of the approximation model and obtain the convergent results with respect to optimal value and optimal solution set. Finally, a so-called stochastic dual dynamic programming approach is proposed to solve the approximate model. Numerical results validate the solution approach of this paper.

Efficient hedging under ambiguity in continuous time

It is well known that the minimal superhedging price of a contingent claim is too high for practical use. In a continuous-time model uncertainty framework, we consider a relaxed hedging criterion based on acceptable shortfall risks. Combining existing aggregation and convex dual representation theorems, we derive duality results for the minimal price on the set of upper semicontinuous discounted claims.

Equi-invariability, bounded invariance complexity and L-stability for control systems

In this paper, we introduce the notions of bounded invariance complexity, bounded invariance complexity in the mean and mean Lyapunov-stability for control systems. Then we characterize these notions by introducing six types of equi-invariability. As a by-product, two new dichotomy theorems for the control system on the control sets are established.

Master‐slave game optimization method of smart energy systems considering the uncertainty of renewable energy

As the uncertainty of renewable energy output brings more and more risks to the day-ahead dispatch of the power grid, an optimization scheduling strategy of a smart energy system based on improved master-slave game model is proposed. Risk factors related to the uncertainties of renewable energy are introduced into the master-slave game model. Taking the smart energy system as the leader and the end users as the follower, an optimized operation model of the smart energy system based on the improved master-slave game model is established, which is transformed into a single-layer linear programming model according to the Karush-Kuhn-Tucher conditions and the duality theorem. The benefits of the system and electric vehicle users in four application scenarios are obtained by the YALMIP algorithm and the sensitivity affecting the economics of the smart energy system is analyzed. The validity of the model is verified by a simulation analysis of actual operation data from the smart energy system in China. The simulation results show that the method proposed in this paper can increase the revenue of the smart energy system by 7%, reduce the risk cost and charging cost of electric vehicle users by 63.92% and 48.34%.

Fock space dualities

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and nuclear physics are discussed. The special case of orthogonal-orthogonal duality is treated in detail. It is shown that, like it was done by Helmers more than half a century ago in the analogous case of symplectic-symplectic duality, one can base a proof of the orthogonal-orthogonal duality theorem and a precise characterization of the relation between the equivalence classes of the dually related irreducible representations on a calculation of characters by combining it with an analysis of the representation of a reflection. Young diagrams for the description of equivalence classes of irreducible representations of orthogonal Lie algebras are introduced. The properties of a reflection of the number non-conserving part in the dual relationship between orthogonal Lie algebras corroborate a picture of an almost perfect symmetry between the partners.