Strong Duality 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.

Low-carbon economic optimization strategy for industrial loads in parks considering source-load-price multivariate uncertainty

A low-carbon economic optimization method is proposed for industrial loads in parks considering multivariate uncertainty. Model of dynamic load node carbon intensity is proposed considering the uncertainty of clean energy and dynamic conventional units carbon intensity based on system currents. The robustness is improved based on multi-interval uncertainty set. The gap is filled by hybrid copula model with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) prices in the uncertainty model. The GARCH is used to built the marginal distribution function of prices, and a hybrid copula model is proposed to obtain the joint probability density function based on the weights calculated by Euclidean distance. The LNCI probability distributions of the regulation potential of loads are proposed with the virtual energy storage and the virtual power model to improve the descriptive ability of uncertainty based on the expansion and contraction functions. Finally, the typical scenarios are extracted based on Monte Carlo and Wasserstein measures. The Column sum Constraint Generation algorithm and strong duality theory are used to get the robust-stochastic optimization. The method proposed in the paper has a positive effect on enterprises to rationalize their production schedules, reduce electricity and carbon emissions costs and increase the revenue from participating in grid regulation.

Extended matrix modeling of integrated energy systems considering network dynamic characteristics and source–load uncertainty

A refined and efficient modeling approach is indispensable for enhancing the timeliness of scheduling strategies for the integrated energy system (IES). Non-flowing variables such as pressure and temperature introduced by energy devices, however, worsen the complexity while improving the completeness for models. Nonlinear and uncertain variables associated with network dynamic characteristics and source–load uncertainty in the systems seriously postpone the modeling processes. An extended matrix modeling approach is proposed in this paper to tackle such issues. New nodes, branches, definition rules and topological matrices are first presented to create an extended matrix model that is compatible with both flowing and non-flowing variables. To boost the modeling efficiency, adaptive piecewise linearization (APL) with extended matrix form and modified Taylor expansion are applied to approximate various nonlinear characteristics. The convex approximation based on Kullback–Leibler (KL) divergence and the strong duality theory characterized by Wasserstein distance are adopted to deal with uncertain variables, which adapt to combine with the above matrix model. A linearized matrix modeling framework applicable to highly automated by computers is thus established. The distributionally robust optimization (DRO) scheduling strategy is verified in a standard test system with the proposed modeling. The effect of dynamic characteristics and uncertain factors on the strategy is discussed. The results show that the extended matrix model simulates the response for real physical systems more effectively, which exhibits the better performances in modeling, objectives and scheduling strategies.

Certifiably optimal rotation and pose estimation based on the Cayley map

We present novel, convex relaxations for rotation and pose estimation problems that can a posteriori guarantee global optimality for practical measurement noise levels. Some such relaxations exist in the literature for specific problem setups that assume the matrix von Mises-Fisher distribution (a.k.a., matrix Langevin distribution or chordal distance) for isotropic rotational uncertainty. However, another common way to represent uncertainty for rotations and poses is to define anisotropic noise in the associated Lie algebra. Starting from a noise model based on the Cayley map, we define our estimation problems, convert them to Quadratically Constrained Quadratic Programs (QCQPs), then relax them to Semidefinite Programs (SDPs), which can be solved using standard interior-point optimization methods; global optimality follows from Lagrangian strong duality. We first show how to carry out basic rotation and pose averaging. We then turn to the more complex problem of trajectory estimation, which involves many pose variables with both individual and inter-pose measurements (or motion priors). Our contribution is to formulate SDP relaxations for all these problems based on the Cayley map (including the identification of redundant constraints) and to show them working in practical settings. We hope our results can add to the catalogue of useful estimation problems whose solutions can be a posteriori guaranteed to be globally optimal.

Perturbational duality in optimization under evenly convex assumptions

In this paper, we consider an optimization problem which is embedded into a family of perturbed problems. We associate with it the classical conjugate dual problem and a surrogate dual problem called quasi-dual. For these two pairs of primal–dual problems, we provide necessary and sufficient conditions for zero duality gap, strong duality and converse duality, under evenly convex assumptions involving appropriate sets associated to the data perturbation function. To this aim, we firstly establish some new results for evenly convex and evenly quasiconvex functions. We conclude by applying our results to two remarkable instances of optimization problems.

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.

Neyman-Pearson Multi-class Classification via Cost-sensitive Learning

Most existing classification methods aim to minimize the overall misclassification error rate. However, in applications such as loan default prediction, different types of errors can have varying consequences. To address this asymmetry issue, two popular paradigms have been developed: the Neyman-Pearson (NP) paradigm and the cost-sensitive (CS) paradigm. Previous studies on the NP paradigm have primarily focused on the binary case, while the multi-class NP problem poses a greater challenge due to its unknown feasibility. In this work, we tackle the multi-class NP problem by establishing a connection with the CS problem via strong duality and propose two algorithms. We extend the concept of NP oracle inequalities, crucial in binary classifications, to NP oracle properties in the multi-class context. Our algorithms satisfy these NP oracle properties under certain conditions. Furthermore, we develop practical algorithms to assess the feasibility and strong duality in multi-class NP problems, which can offer practitioners the landscape of a multi-class NP problem with various target error levels. Simulations and real data studies validate the effectiveness of our algorithms. To our knowledge, this is the first study to address the multi-class NP problem with theoretical guarantees. The proposed algorithms have been implemented in the R package npcs, which is available on CRAN.

Wolfe type duality on quasidifferentiable mathematical programs with vanishing constraints

This article is devoted to the study of duality results for optimization problems with vanishing constraints in nonsmooth case. We formulate Wolfe type dual and establish weak, strong, converse, restricted converse and strict converse duality results for mathematical programs with vanishing constraints involving quasidifferentiable functions. Under the assumption of invex and strictly invex functions with respect to a convex compact set.

Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating to the primal problem, NIMSIPVC, and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.

Solving a bilevel strategic offering model for a generation company in a hydrothermal energy market: A new convexification approach

This paper proposes a bilevel strategic offering (SO) model suitable for a strategic company (SC) that operates in a hydro-dominated day-ahead energy market. The upper-level model of the proposed SO model maximizes the SC’s profits, while the lower-level model embodies a hydrothermal market clearing (MC). This MC incorporates detailed hydro and thermal constraints for units of both the SC and non-strategic companies (NSC), in addition to the network constraints. The proposed solution approach entails replacing the lower-level model with its corresponding primal and dual constraints, along with a strong duality equality constraint. This leads to a non-convex mathematical programming with equilibrium constraints (MPEC) model, which is then reformulated to feature only a single bilinear term in its structure. Finally, the recast MPEC model is convexified around an initial point and solved iteratively until a convergence criteria is achieved. The model and solution approach proposed are evaluated on an adapted version of the IEEE 24-bus system, exploring scenarios where the SC acts strategically or non-strategically (perfect competition).

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

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

An interactive framework for strategic participation of a price-maker energy hub in the local gas and power markets based on the MPEC method

Power-to-gas (P2G) technologies within smart energy hubs (SEH) offer unprecedented operational flexibility to distribution networks. This is achieved by integrating various energy carriers both regionally and locally, particularly in the presence of renewable energy sources (RES). According to this subject, the present work outlines an interactive risk-based model and meticulous mathematical solution to justifiably incorporate the P2G units into the SEH to participate in the local electricity and natural gas markets strategically. As a price-maker prosumer, the SEH utilizes cross-product arbitrage strategies to optimize its proposals for power and gas exchanges in the local markets. The participation model for SEH is developed as a bi-level framework. At the top level, the goal is to maximize SEH's profit while accounting for the risks posed by RES. At the bottom level, the aim is to increase social welfare under network constraints through local market settlements. Finally, a mathematical program with equilibrium constraints (MPEC) is offered as a solution to solve the bi-level model. Using integer disjunctions on the complementarity and slackness criteria and the strong duality theory on the bilinear product terms, a mixed-integer linear programming (MILP) has been derived to approximate the suggested MPEC formulation. It can be concluded from the results that the integration of P2G technologies and the use of mutual product arbitrage in the local electricity and natural gas markets, not only increases the profit of the SEH, but also reduces operating costs and improves social welfare as well as cost-recovery of the entire network.

Stochastic linear–quadratic control problems with affine constraints

This paper investigates the stochastic linear–quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality theory, both the dual problem and the state feedback form of the solution are obtained for the primal problem. Under the Slater condition, the strong duality is proved between the dual problem and the primal problem, and the KKT condition is also provided for solving the primal problem. Moreover, a new sufficient condition is given for the invertibility assumption, which ensures the uniqueness of the solutions to the dual problem. Finally, two numerical examples are provided to illustrate our main results.

Lossless convexification and duality

The main goal of this paper is to investigate the strong duality of non-convex semidefinite programming problems (SDPs). In the optimization community, it is well-known that a convex optimization problem satisfies strong duality if Slater’s condition holds. However, this result cannot be directly generalized to non-convex problems. In this paper, we prove that a class of non-convex SDPs with special structures satisfies strong duality under Slater’s condition. Such a class of SDPs arises in SDP-based control analysis and design approaches. Throughout the paper, several examples are given to support the proposed results. We expect that the proposed analysis can potentially deepen our understanding of non-convex SDPs arising in the control community and promote their analysis based on KKT conditions.

A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

Wasserstein distributionally robust optimization has emerged as a recent topic with broader applications in operations research and machine learning. Various proofs have been presented in the literature, each differing in assumptions and levels of generality. In “A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization,” Zhang, Yang, and Gao present a novel elementary proof that not only shortens existing frameworks but also offers surprising generalizations. Leveraging classical Legendre—Fenchel duality, they demonstrate that strong duality is contingent on a certain interchangeability principle. Moreover, they extend this duality result to encompass risk-averse optimization and globalized distributionally robust counterparts.

Distributionally robust risk-resistant synergistic restoration strategy for integrated electricity and gas system with high-fidelity gas dynamic modeling

A synergistic restoration strategy offers a promising solution for expediting restoration of integrated electricity and gas system (IEGS) following blackouts, particularly under the increasing energy interdependence between these two subsystems. However, achieving synergistic restoration is challenging in 1) distinct operational characteristics between electric and gas flow rates, leading to difficulties in accurately modeling the restoration processes with significantly varying system states; 2) potential risks posed by uncertainties, such as fluctuating renewable energy and random contingencies, undermining the effectiveness of restoration strategy. To address these challenges, this paper proposes a novel distributionally robust risk-resistant synergistic restoration strategy for IEGS. Firstly, to accurately capture the dynamics of slow gas flow relative to instantaneous power flow, we propose a high-fidelity linear dynamic gas flow model with multiparametric disaggregation technique, demonstrating satisfactory accuracy in restoration scenarios. Then, the risks by uncertainties of renewable energy and contingencies are handled within distributionally robust framework. Moreover, the IEGS restoration problem is reformulated as a two-stage robust optimization problem by convex conservative approximation and strong duality theory, which is solved by a nested column-and-constraint generation algorithm. Finally, simulation results validate the effectiveness of the proposed restoration strategy, showing the improved restoration efficiency, security, and risk-resistant performance.

A note on duality in nonsmooth minimax fractional optimization problems with an infinite number of constraints

In this paper, we introduce duality for nonsmooth minimax fractional semi-infinite optimization problems in Asplund space by using some advanced tools of variational analysis and generalized differentiation. We propose a dual problem to the primal one; then weak, strong and converse-like duality relations are explored. In addition, some of these results are applied to multiobjective optimization problems.

Optimal scheduling of a community Multi-Energy system in energy and flexible ramp markets considering Vector-Coupling storage Devices: A hybrid Fuzzy-IGDT/Stochastic/Robust optimization framework

Community multi-energy systems (CMESs) are emerging as a pivotal solution to address the pressing issues of the current energy crisis and environmental pollution. This paper introduces a comprehensive multi-horizon risk-averse scheduling strategy for CMESs, targeting participation in energy and flexible ramp markets while meeting the demands of electric, natural gas, and hydrogen vehicle charging stations, data centers, and residential and commercial buildings. The proposed CMES incorporates power-to-x vector coupling storage (PtX-VCS) systems, enabling the storage of electricity in various forms to cater to multi-energy demands. Additionally, an integrated demand response (IDR) program, incorporating incentive and load-shifting strategies, is applied to enhance operational efficiency and system flexibility. To manage uncertainties effectively, a hybrid stochastic/robust/fuzzy information gap decision theory (IGDT) method is employed. This approach, formulated as a tri-level optimization model, is transformed into a single-level model using strong duality theory. The results demonstrate a notable reduction in daily operational costs by approximately 5% and 4% through the implementation of the IDR program and PtX-VCS technologies, respectively. Moreover, the CMES’s potential to participate in the flexible ramping market leads to an approximately 7% reduction in total operational costs.

Integrating Risk-Averse and Constrained Reinforcement Learning for Robust Decision-Making in High-Stakes Scenarios

This paper considers a risk-averse Markov decision process (MDP) with non-risk constraints as a dynamic optimization framework to ensure robustness against unfavorable outcomes in high-stakes sequential decision-making situations such as disaster response. In this regard, strong duality is proved while making no assumptions on the problem’s convexity. This is necessary for some real-world issues, e.g., in the case of deprivation costs in the context of disaster relief, where convexity cannot be ensured. Our theoretical results imply that the problem can be exactly solved in a dual domain where it becomes convex. Based on our duality results, an augmented Lagrangian-based constraint handling mechanism is also developed for risk-averse reinforcement learning algorithms. The mechanism is proved to be theoretically convergent. Finally, we have also empirically established the convergence of the mechanism using a multi-stage disaster response relief allocation problem while using a fixed negative reward scheme as a benchmark.

Optimizing local electricity markets: A bi-level primal-dual approach for integrating price-based demand response and distribution locational marginal pricing

This paper addresses the conundrum of optimizing electricity consumption patterns in response to fluctuations in demand and price, a task managed by both load aggregators and the distribution system operator (DSO). The conventional approaches in the literature to integrate demand response (DR) into optimal power flow (OPF) problems typically overlook the price responsiveness of consumers or simplify power flow equations to account for price-elastic demand.In this paper, we strive to close this gap by introducing a bi-level primal-dual optimization framework that incorporates aggregators' objectives into the market-clearing process while preserving the precision of the OPF equations. At the upper level, the model seeks to minimize both the total payment and peak load. The lower level, structured as a second-order conic (SOC) problem, aims to reduce overall generation costs subject to constraints of the second-order conic (SOC) branch flow model (BFM). The two levels interact through the optimal demand response and distribution locational marginal prices (DLMPs). The principles of convex duality principles together with the strong duality constraint are leveraged to transform the market-clearing problem into a single primal-dual problem. We also circumvent the non-linearity that stems from the DR payment term by incorporating discretizing loads and the big-M method, thus converting the problem into a mixed-integer SOC (MI-SOC) formulation. The merits of the proposed MI-SOC framework are validated through case studies conducted on the IEEE 33-bus test system, showcasing its potential to enforce price-elasticity demand response in distribution system's electricity markets.