Strong Duality Theorems Research Articles

Robust Semi-Infinite Interval Equilibrium Problem Involving Data Uncertainty: Optimality Conditions and Duality

In this paper, we model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. We particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending the results from the literature. We introduce the dual robust version of the above problem, prove the Mond–Weir-type weak and strong duality theorems, and illustrate our results with an example.

A decision-making model for joint energy and reserve scheduling of wind power producers with local intraday demand response exchange market

In this paper, a three-stage stochastic bi-level optimization framework is presented for optimal participation of wind power producers (WPPs) in day-ahead (DA), intraday, and balancing markets. In this framework, to leverage demand response (DR) services, a peer-to-peer (P2P) energy trading platform is implemented that allows local load aggregators (LAs) to contribute to the intraday markets improving both LAs’ and WPP’s benefits. Participating in the intraday DR exchange (IDRX) market enables WPP to purchase DR services from LAs, to reduce the penalty cost on the deviation between the day-head bidding and the real-time dispatch. A Stackelberg game for the bi-level decision-making model captures the conflict of interests between the WPP and LAs, in which, the upper level seeks to maximize WPP profit, while the lower level aims to maximize LAs’ economic surplus. The bi-level model is converted into its equivalent single-level mixed-integer quadratic problem (MIQP) employing the Karush-Kuhn-Tucker (KKT) conditions and strong duality theorem. Simulation results show that participation of the WPP in the IDRX market and employing spinning reserve and DR services for compensating the uncertainties are greatly dependent on its risk preferences and increase its expected profit in all conditions, significantly.

Refinements and Extensions of Some Strong Duality Theorems in Conic Linear Programming

Refinements and Extensions of Some Strong Duality Theorems in Conic Linear Programming

Research on defense strategies for power system frequency stability under false data injection attacks

The rapid development of cyber-physical power system has highlighted the pressing issue of cyber attacks (CAs) faced by large-scale renewable energy power systems (LREPS). The injection of false data attacks on critical electrical equipment by attackers can lead to abnormal system frequencies, triggering cascade failures and causing widespread blackouts. This paper introduces a tri-level defense model specifically crafted for LREPS, comprising an upper, middle, and lower level (defender-attacker-operator). By promoting collaborative decision-making among these agents, the model aims to mitigate the rate of change of frequency (RoCoF) and effectively withstand CAs. The upper-level employs an economic-inertia optimization strategy, scheduling generator units to maximize system inertia while minimizing generation costs. By leveraging the strong duality theorem, the tri-level model is decomposed into a bi-level model consisting of a master problem and a subproblem, solved using the Benders decomposition method. Furthermore, this paper also considers augmenting the virtual inertia of renewable energy units when defense resources are insufficient, ensuring RoCoF remains within limits after CAs. The effectiveness and superiority of the proposed model and strategy are validated through case studies based on the improved IEEE 118-bus system. The results demonstrate that the model can reduce load shedding caused by the worst-case CAs, enhance system inertia, lower RoCoF, and improve the stability of LREPS.

Coordinated risk-averse distributionally robust optimization for maintenance and generation schedules of offshore wind farm cluster

The coordinated optimization of maintenance and generation schedules of multiple offshore wind farms (OWFs) bring great benefits in reducing the maintenance and operation cost of OWF cluster. Firstly, a two-stage coordinated risk-averse distributionally robust optimization (CRDRO) model for maintenance and generation schedules of wind turbines (WTs) in OWF cluster is proposed. This model takes into account uncertainties related to the prediction errors in WTs’ maximum available active power outputs, the travel time of maintenance vessels, and the high operational cost risks associated with these uncertainties. A feasible route generating algorithm is designed to obtain all feasible maintenance combinations for WTs and the corresponding routes of maintenance vessels for the CRDRO model. The risk-averse term, which is a weighted sum of expected operation costs and GlueVaR value, is used to measure the tail risk of the worst probability distribution. To solve the two-stage CRDRO model, the risk-averse term is simplified into the form of expected value calculation, which can be further transformed into the tractable form by using the strong duality theorem. Finally, the two-stage CRDRO model is solved by the column-and-constraint generation algorithm, with the nonconvex sub-problem being addressed by the outer approximation algorithm. Case studies on an real-world cluster comprising three OWFs verify the correctness and efficiency of the proposed method, and demonstrate that the CRDRO model enhances the overall economic efficiency and reduces the high operational cost risk under extreme scenarios.

Data-driven two-stage robust optimization dispatching model and benefit allocation strategy for a novel virtual power plant considering carbon-green certificate equivalence conversion mechanism

Aiming at a large number of decentralized resources such as rural biomass, rooftop photovoltaic, and decentralized wind power, a novel rural virtual power plant (VPP) has been conceptualized. The VPP is integrated by biomass waste conversion system (BWC), carbon to energy (C2P), demand response aggregator (DRA), and distributed renewable energy (DRE), namely, BDCD-VPP. Then, this paper proposes a data-driven two-stage robust optimization dispatching model for BDCD-VPP considering carbon-green certificates equivalent conversion mechanism. This model aims to address the uncertainty of wind power plants (WPP) and photovoltaic (PV) power generation. Strong duality theorem (SDT) and C&CG algorithm are applied to construct this model. Thirdly, the Aumann-Shapley method (A-S) was enhanced by incorporating risk factor, cost factor, and carbon reduction factor. This refinement results in the development of a synergistic scheduling revenue allocation method for electricity‑carbon-green certificates. Finally, the Lankao rural energy revolution pilot program in China is selected as the case study, the results showed: (1) The BDCD-VPP aggregates distributed energy sources such as the rural WPP and the PV to realize electricity‑carbon-electricity cycle effect. The BDCD-VPP generates a green certificate revenue of 47.17¥/MWh and exhibited carbon emissions of 0.37 t/MWh. The proposed Carbon-Green Certificates Equivalent Conversion Mechanism increases BDCD-VPP benefit ratio by 3.52%. (2) The proposed two-stage robust optimization dispatching model enhances the ability of BDCD-VPP to adapt to uncertainty. Compared to the day-ahead stage, biomass power generation (BPG) and waste power generation (WPG) increase upward peaking power by 24.05% and 9.99% in intra-day stage. Flue gas treatment system (FG-TS), gas-power plant carbon capture (GPPCC) and power to gas (P2G) increase downward peaking power by 65.15%, 70.99% and 25.94%. (3) Utilizing the proposed benefit allocation methodology, BPG and WPG need to concede 38.25¥/MWh and 65.02¥/MWh for carbon emissions associated with electricity. WPP and PV need to concede 64.25¥/MWh and 33.71¥/MWh for power generation uncertainty. GPPCC, P2G, DRA, and small hydropower station (SHS) obtain revenues of 24.48 ¥/MWh, 55.07 ¥/MWh, 70.38 ¥/MWh, and 51.30 ¥/MWh. Compared to the A-S, the proposed benefit allocation methodology exhibits 6.92% improvement in satisfaction. Overall, the proposed data-driven two-stage robust optimization dispatching model and benefit allocation strategy facilitates the optimal aggregation and utilization of rural distributed energy resources, considering the interests of various stakeholders. This is conducive to achieving a clean and low-carbon transformation of the overall energy structure.

Optimal bidding strategy of a wind power producer in Chinese spot market considering green certificate trading.

In China, wind power producers will participate in the spot market as strategic producers. They should submit offering prices and forecasted production to the independent system operator. Intraprovincial and interprovincial green certificate trading, as a mechanism to promote the development of wind power, is advanced in parallel with the spot market. Studying the bidding strategies of wind power producers in the spot market, especially with the introduction of intraprovincial and interprovincial green certificate trading, has great practical significance for the stable operation of wind power producers and the construction of a renewable energy-friendly electricity market. This paper proposes a bi-level stochastic model to simulate the bidding strategy of a wind power producer in the spot market. The upper level problem of the model describes the decision-making process of the wind power producer, and the lower level problems include the clearing process of the spot market and the TGC market, in which the spot market includes the day-ahead market and the balance market, and the TGC market includes the intraprovincial and interprovincial TGC markets. Then, using Karush-Kuhn-Tucker (KKT) conditions, the strong duality theorem, and the big M method, the bi-level model is recast into a mixed integer linear (MIL) model convenient to be solved. Finally, an example is given to analyze the wind power producer's optimal bidding strategy under different electricity market environments and different wind power penetration levels.

Lagrangian dual method for solving stochastic linear quadratic optimal control problems with terminal state constraints

A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost functional and a surjectivity condition on the linear constraint mapping. Based on the Lagrangian duality, two approaches are proposed to solve the constrained stochastic LQ problem. First, a theoretical method is given to construct the closed-form solution by the strong duality. Second, an iterative algorithm, called augmented Lagrangian method (ALM), is proposed. The strong convergence of the iterative sequence generated by ALM is proved. In addition, some sufficient conditions for the surjectivity of the linear constraint mapping are obtained.

A Stackelberg Game Based Optimization Method for Heterogeneous Building Aggregations in Local Energy Markets

This paper proposes an optimal operational method and a customized pricing strategy for a Microgrid (MG) incorporating heterogeneous buildings from different communities in a local energy market (LEM). A customized pricing strategy in the LEM for the heterogeneous building aggregations (BAs) is proposed considering their heterogeneous insulation performances. In this regard, a hierarchical framework with two levels for optimal decision-making of the MG as the price maker and BAs as the price takers in the LEM are presented. At the upper level, the MG aims to maximize its profit by optimizing the customized prices and its energy schedules. At the lower level, a detailed physical model of the building with adjustable heating, ventilation, and air conditioning (HVAC) systems is developed while considering the building's thermal dynamics. The objective of each BA in each community is to minimize its operating cost while ensuring the users' thermal comfort according to MG's prices. The Stackelberg game is used to model the interactions between the MG (leader) as the price maker and multiple heterogeneous BAs (followers) as the price takers in the LEM. Then, the proposed optimization model is converted into a mixed integer linear programming using the Karush-Kuhn-Tucker conditions and strong duality theorem. Numerical results show that the hierarchical method benefits both the MG and heterogeneous BAs with customized pricing schemes. The HVAC of the building with good thermal insulation performance can make more adjustments to indoor temperature to reduce its operating costs. Moreover, it has a more significant impact on MG's electricity prices.

Modeling Strategic Behaviors for GenCo With Joint Consideration on Electricity and Carbon Markets

With the development of carbon markets, conventional generators are undergoing electricity markets and carbon markets at the same time. To fulfill the carbon market requirements and cope with the emerging cross-market arbitrage, the multi-market strategic behaviors of the generation company (GenCo) should be carefully studied. In this paper, the strategic behaviors of a GenCo with joint consideration of the electricity market and carbon market are modeled. First, the dynamic carbon emission intensity (CEI) is introduced to achieve the numerical relationship between the power output and carbon emissions, by which GenCo is demonstrated as a prosumer in the carbon market. Then, the strategy model is built in a bi-level structure. GenCo's multi-market profits maximization model is built at the upper level to determine the best bidding curves. The lower level is made up of two market-clearing models: the day-ahead electricity market for economic power dispatch and the bi-directional carbon market accounting for prosumers' variable trading positions. Then, the high dimensional nonlinearity bilevel model is reformulated and part-linearized by the Karush-Kuhn-Tucker and strong duality theorems. Case studies based on an IEEE 30-bus system indicate that dynamic CEIs have a significant impact on GenCo's cross-market arbitrage by determining its trading volume and positions in the carbon market.

Constraint Qualifications for Vector Optimization Problems in Real Topological Spaces

In this paper, we introduce a series of definitions of generalized affine functions for vector-valued functions by use of “linear set”. We prove that our generalized affine functions have some similar properties to generalized convex functions. We present examples to show that our generalized affinenesses are different from one another, and also provide an example to show that our definition of presubaffinelikeness is non-trivial; presubaffinelikeness is the weakest generalized affineness introduced in this article. We work with optimization problems that are defined and taking values in linear topological spaces. We devote to the study of constraint qualifications, and derive some optimality conditions as well as a strong duality theorem. Our optimization problems have inequality constraints, equality constraints, and abstract constraints; our inequality constraints are generalized convex functions and equality constraints are generalized affine functions.

Zero-Sum Games and Linear Programming Duality

The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig is known to be incomplete. We explain and combine classical theorems about solving linear equations with nonnegative variables to give a correct alternative proof more directly than Adler. We also extend Dantzig’s game so that any max-min strategy gives either an optimal LP solution or shows that none exists.

Second-Order Optimality Conditions and Duality for Multiobjective Semi-Infinite Programming Problems on Hadamard Manifolds

This paper is devoted to the study of multiobjective semi-infinite programming problems on Hadamard manifolds. We consider a class of multiobjective semi-infinite programming problems (abbreviated as MSIP) on Hadamard manifolds. We use the concepts of second-order Karush–Kuhn–Tucker stationary point and second-order Karush–Kuhn–Tucker geodesic pseudoconvexity of the considered problem to derive necessary and sufficient second-order conditions of efficiency, weak efficiency and proper efficiency for MSIP along with certain generalized geodesic convexity assumptions. Moreover, we formulate the second-order Mond–Weir-type dual problem related to MSIP and deduce weak and strong duality theorems relating MSIP and the dual problem. The significance of our results is demonstrated with the help of non-trivial examples. To the best of our knowledge, this is the first time that second-order optimality conditions for MSIP have been studied in Hadamard manifold setting.

A data-driven Stackelberg game approach applied to analysis of strategic bidding for distributed energy resource aggregator in electricity markets

In the context of the participation of distributed energy resource (DER) aggregators in day-ahead energy and reserve markets, most of the existing analysis models of strategic bidding behavior of DER aggregators ignore the consideration of the safe operation of distribution systems. To this end, this paper builds a one-leader multi-follower Stackelberg game model considering the safety checking of distribution systems with the discrete features of capacitor banks and tap ratio of the substation transformer in the lower level. Mathematically, this model constitutes a bi-level mixed-integer nonlinear programming (BMINLP) problem. We convert the BMINLP model to a bi-level mixed-integer linear programming (BMILP) model based on KKT conditions, strong duality theorem, and big-M approach, which is still difficult to be solved directly. As a result, we propose a customized data-driven algorithm based on sampling without tuning hyper-parameters and an improved linear surrogate function method, to accurately fit the corresponding constraints of the lower-level model of BMILP problem, and then transform the BMILP problem into a single-level mixed-integer linear programming (MILP) problem to solve. The simulation of data-driven algorithm and comparison with the particle swarm optimization algorithm shows the high practicability of the bi-level Stackelberg game model and the effectiveness of the data-driven algorithm.

Nonsmooth semi-infinite roughly B-invex multi-objective programming problems

In this paper, we will study a nonsmooth semi-infinite multi-objective programming problem involving roughly B-invex functions. Sufficient optimality conditions for the primal problem are derived. We shall formulate Mond–Weir type dual and establish weak and strong duality theorems under various roughly B-invexity assumptions.

The KKT Optimality Conditions and Duality for Constrained Programming Problem with Generalized α- Convex Fuzzy Functions

This paper mainly studies the mixed constraint interval programming problem under the generalized convex fuzzy mapping. Firstly, this paper give the concepts of fuzzy mappings, such as quasiconvex, strictly quasiconvex, pseudoconvex and strictly pseudoconvex. Then, the relation of generalized convex fuzzy mapping is studied and some properties are obtained. Finally, the necessary and sufficient KKT conditions are given, and the duality problem is established. The weak duality, strong duality and inverse duality theorem of fuzzy interval programming are proved.

Second-Order Symmetric Duality for Multiple Objectives Nonlinear Programming Under Generalizations of Cone-Preinvexity Functions

In this article, we present generalizations of the cone-preinvexity functions and study a pair of second-order symmetric solutions for multiple objective nonlinear programming problems under these generalizations of the cone-preinvexity functions. In addition, we establish and prove the theorems of weak duality, strong duality, strict converse duality, and self-duality by assuming the skew-symmetric functions under these generalizations of the cone-preinvexity functions. Finally, we provide four nontrivial numerical examples to demonstrate that the results of the weak and strong duality theorems are true.

A bi-level approach for participation of hybrid transmission operating companies in the day-ahead market, considering energy storage systems

This paper presents a new approach for participation strategy of Hybrid Transmission Operating Companies (HTOCs) in the day-ahead energy market. In this paper HTOC refers to private companies that have won physical transmission rights (PTR) and also own energy storage facilities. In the introduced hybrid structure, the HTOC can make profit in two different ways; first, through the flow of transmission lines (the MW-Mile transmission pricing method is used here); and second, through an optimum scheduling of its energy storage system. The optimum charge/discharge pattern of energy storage systems, evaluated in this paper, would also increase the MW-Mile profit of the company. The proposed method is a bi-level optimization model. In the upper level the profit of the HTOC is maximized and in the lower level the market clearing process of the independent system operator (ISO) has been modeled. The uncertain behavior of rival units is modeled using robust optimization (RO) method. The Karush-Kuhn-Tucker (KKT) optimality conditions are implemented to convert the proposed bi-level problem into its equivalent single-level problem. The Big-M method and the strong duality theorem (SDT) are used to linearize the proposed method. To demonstrate the performance of the introduced structure for HTOC, the proposed strategy has been implemented on a standard IEEE 9-bus and 57-bus test systems. The results show that HTOC can achieve a maximum guaranteed profit using storage resource (SR). It is also concluded that the presence of SR has not only increased HTOC's profit but also helped to make the structure more flexible against the uncertain parameters. In addition, the RO as a calculation risk method has been perfectly matched with the implementation of SR as a source of hedging uncertainties in managing the risk.

Semi-infinite Mathematical Programming Problems involving Generalized Convexity

In this paper, we consider semi-infinite mathematical programming problems withequilibrium constraints (SIMPEC). By using the notion of convexificators, we establish sufficient optimality conditions for the SIMPEC. We formulate Wolfe and Mond-Weir type dual models for the SIMPEC under generalized convexity assumptions. Moreover, weak and strong duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators.

Approximate optimality conditions and mixed type duality for semi-infinite multiobjective programming problems involving tangential subdifferentials

The aim of this paper is to study the approximate solution for semi-infinite multiobjective programming problems related to tangential subdifferentials. Under some mild assumptions, we obtain approximate necessary and sufficient optimality conditions of a (weakly) quasi $ \varepsilon $-efficient solution for semi-infinite multiobjective programming problems. We also introduce a mixed type dual model to the semi-infinite multiobjective programming problem, which is different from the one considered in the literature. Finally, we derive some approximate weak, strong and converse duality theorems under the assumptions of generalized convexity. Our results extend and improve the corresponding results in the literature. Some examples are given to illustrate some advantages of our results.