Primal-dual Formulation Research Articles

A relaxation viewpoint to Unbalanced Optimal Transport: Duality, optimality and Monge formulation

We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex envelope of a cost for non-negative Dirac masses. New general primal-dual formulations, optimality conditions, and metric-topological properties are carefully studied and discussed.

Optimal strategies for large-scale pursuers against one evader: A mean field game-based hierarchical control approach

This paper proposes a two-level hierarchical control approach based on pursuit-evasion game and mean field game for the problem of large-scale pursuers with multi-population against a single evader, which implements that the evader is surrounded by pursuers. At the upper layer, we model the pursuit-evasion game between the centers of the pursuer populations and the single evader, which is formulated as a linear quadratic differential game (LQDG) to obtain the optimal control of each player. Then the optimal trajectories derived from the optimal controls are input to the lower layer as the reference trajectories. At the lower layer, we formulate the tracking of reference trajectories and terminal surrounding to the evader of large-scale pursuers with multi-population as a multi-population mean-field game (MPMFG), which solves the communication and computing difficulties caused by large-scale agents. Then, we derive the variational primal–dual formulation of the proposed MPMFG model and solve it with CA-Net, a coupled alternating neural network approach. Finally, simulation experiments are performed under various pursuit-evasion scenarios, and it is verified that the proposed game-based two-level hierarchical control approach is feasible and effective.

On duality principles and related convex dual formulations suitable for local and global non-convex variational optimization

Abstract This article develops duality principles, a related convex dual formulation and primal dual formulations suitable for the local and global optimization of non convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg–Landau type equation. Other applications include primal dual variational formulations for a Burger’s type equation and a Navier–Stokes system. We emphasize the novelty here is that the first dual variational formulation developed is convex for a primal formulation which is originally non-convex. Finally, we also highlight the primal dual variational formulations presented have a large region of convexity around any of their critical points.

A necessary condition for extremality of solutions to autonomous obstacle problems with general growth

Let us consider the autonomous obstacle problem minv∫ΩF(Dv(x))dxon a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity. Our aim is to find a necessary condition for the extremality of the solution, which exists and it is unique, thanks to a primal–dual formulation of the problem. The proof is based on classical arguments of Convex Analysis and on Calculus of Variations’ techniques.

Investing in Wind Energy Using Bi-Level Linear Fractional Programming

Investing in wind energy is a tool to reduce greenhouse gas emissions without negatively impacting the environment to accelerate progress towards global net zero. The objective of this study is to present a methodology for efficiently solving the wind energy investment problem, which aims to identify an optimal wind farm placement and capacity based on fractional programming (FP). This study adopts a bi-level approach whereby a private price-taker investor seeks to maximize its profit at the upper level. Given the optimal placement and capacity of the wind farm, the lower level aims to optimize a fractional objective function defined as the ratio of total generation cost to total wind power output. To solve this problem, the Charnes-Cooper transformation is applied to reformulate the initial bi-level problem with a fractional objective function in the lower-level problem as a bi-level problem with a fractional objective function in the upper-level problem. Afterward, using the primal-dual formulation, a single-level linear FP model is created, which can be solved via a sequence of mixed-integer linear programming (MILP). The presented technique is implemented on the IEEE 118-bus power system, where the results show the model can achieve the best performance in terms of wind power output.

Learning Equilibria in Matching Markets with Bandit Feedback

Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) are, unfortunately, of limited value in the learning setting, given that preferences are inherently uncertain and destabilizing while they are being learned. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that “optimism in the face of uncertainty,” the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.

Revealing Vulnerability of N-1 Secure Power Systems to Coordinated Cyber-Physical Attacks

Coordinated cyber-physical (CCP) attacks have attracted wide attention in power systems because of their potential to cause severe disturbance including cascading failures. However, as realistic power systems operate within the N-1 security criterion, existing CCP attacks may be in part mitigated. As such, this paper analyzes impacts of CCP attacks on the vulnerability of the power systems that employ the N-1 security constrained optimal power flow (SCOPF). Specifically, a tri-level model is proposed to analyze the CCP attack impacts, whereby the adversary coordinates a physical attack with cyber attacks to initiate and propagate post-contingency overload based on N-1 SCOPF. A methodology utilizing semidefinite programming (SDP) relaxation and primal-dual formulation is proposed to transform the tri-level model into a conic optimization, such that the model can be easily solved by SDP solvers. Case studies on the IEEE 14, 57, and 118 bus test systems demonstrate that the CCP attacks in power systems with N-1 security criterion are able to cause N-1-1 contingency and even trigger cascading failures.

Convergence Results for Primal-Dual Algorithms in the Presence of Adjoint Mismatch

Most optimization problems arising in imaging science involve high-dimensional linear operators and their adjoints. In the implementations of these operators, changes may be introduced for various practical considerations (e.g., memory limitation, computational cost, convergence speed), leading to an adjoint mismatch. This occurs for the X-ray tomographic inverse problems found in computed tomography (CT), where a surrogate operator often replaces the adjoint of the measurement operator (called the projector). The resulting adjoint mismatch can jeopardize the convergence properties of iterative schemes used for image recovery. In this paper, we study the theoretical behavior of a panel of primal-dual proximal algorithms, which rely on forward-backward-(forward) splitting schemes when an adjoint mismatch occurs. We analyze these algorithms by focusing on the resolution of possibly nonsmooth convex penalized minimization problems in an infinite-dimensional setting. Using tools from fixed point theory, we show that they can solve monotone inclusions beyond minimization problems. Such findings indicate that these algorithms can be seen as a generalization of classical primal-dual formulations. The applicability of our findings is also demonstrated through two numerical experiments in the context of CT image reconstruction.

A Joint Optimization Model for Transmission Capacity and Wind Power Investment Considering System Security

Due to continuous growth in electric demand and increasing connection of renewable energy, the power systems are being operated with smaller stability margins. Therefore, a sufficient loading margin is essential to maintain the system secure and ensure voltage stability. To this end, this paper studies the joint optimization of transmission capacity and wind power investment problem, the unique feature of which is incorporating voltage stability margin (VSM) in the planning model. A bi-level model has been formulated whose upper level minimizes the total investment and operation cost minus the weighted VSM. The lower level evaluates the VSM given the optimal expansion plan from the upper level. In addition, the stochastic nature of wind power and load can impact voltage stability. Thus, uncertainties related to intermittent wind generation and demand must be modeled. We use an approximated linear representation to model the AC power system at both levels of the problem. The duality theory (primal-dual formulation) is utilized to transform bi-level programming into single-level mathematical programming. The validity of the constructed methodology is demonstrated on the IEEE 24-bus RTS, which indicates the efficacy and feasibility of the presented model.

Unit commitment with frequency‐arresting adequacy constraint: Modelling and pricing aspects

Frequency-arresting (FA) adequacy has received an increasing attention in operation of power systems with intermittent, electronically connected wind and sfolar generation. Presence of high penetration of renewable energy leads to reduced amount of inertia in the power system and the need for more sustainable market design. This paper addresses the FA adequacy by integrating Nadir and rate of change of frequency constraints in unit commitment formulation. Here, a mixed-integer linear program (MILP) is proposed for addressing the FA adequacy. The proposed MILP model of FA adequacy is non-convex and accordingly, FA adequacy pricing requires non-convex pricing techniques. The explicit and implicit uplift payment techniques are proposed and implemented for FA adequacy pricing using the authors' proposed MILP model. The primal–dual formulation is employed for implicit uplift technique. The proposed MILP model of FA adequacy along with its pricing techniques is carefully studied through an illustrative example system. For further discussions, the Nordic 44-node test network is studied.

Faster first-order primal-dual methods for linear programming using restarts and sharpness

First-order primal-dual methods are appealing for their low memory overhead, fast iterations, and effective parallelization. However, they are often slow at finding high accuracy solutions, which creates a barrier to their use in traditional linear programming (LP) applications. This paper exploits the sharpness of primal-dual formulations of LP instances to achieve linear convergence using restarts in a general setting that applies to alternating direction method of multipliers (ADMM), primal-dual hybrid gradient method (PDHG) and extragradient method (EGM). In the special case of PDHG, without restarts we show an iteration count lower bound of \(\Omega (\kappa ^2 \log (1/\epsilon ))\), while with restarts we show an iteration count upper bound of \(O(\kappa \log (1/\epsilon ))\), where \(\kappa \) is a condition number and \(\epsilon \) is the desired accuracy. Moreover, the upper bound is optimal for a wide class of primal-dual methods, and applies to the strictly more general class of sharp primal-dual problems. We develop an adaptive restart scheme and verify that restarts significantly improve the ability of PDHG, EGM, and ADMM to find high accuracy solutions to LP problems.

Distributionally Robust Linear and Discrete Optimization with Marginals

In optimization problems, decisions are often made in the face of uncertainty that might arise in the form of random costs or benefits. In “Distributionally Robust Linear and Discrete Optimization with Marginals,” Louis Chen, Will Ma, Karthik Natarajan, David Simchi-Levi, and Zhenzhen Yan study a robust bound of linear and discrete optimization problems in which the objective coefficients are random and the set of admissible joint distributions is assumed to be specified only up to the marginals. They provide a primal-dual formulation for this problem, and in the process, unify existing results with new results. They establish NP-hardness of computing the bound for general polytopes and identify two sufficient conditions—one based on a dual formulation and one based on sublattices that provide a class of polytopes where the robust bounds are efficiently computable.

A Primal–Dual Learning Algorithm for Personalized Dynamic Pricing with an Inventory Constraint

We consider the problem of a firm seeking to use personalized pricing to sell an exogenously given stock of a product over a finite selling horizon to different consumer types. We assume that the type of an arriving consumer can be observed, but the demand function associated with each type is initially unknown. The firm sets personalized prices dynamically for each type and attempts to maximize the revenue over the season. We provide a learning algorithm that is near optimal when the demand and capacity scale in proportion. The algorithm utilizes the primal–dual formulation of the problem and learns the dual optimal solution explicitly. It allows the algorithm to overcome the curse of dimensionality (the rate of regret is independent of the number of types) and sheds light on novel algorithmic designs for learning problems with resource constraints.

On the solution of robust transmission expansion planning using duality theorem under polyhedral uncertainty set

The solution techniques of tri-level robust optimization for large-scale power systems continue to remain as a great challenge. Existing techniques rely on using either Karush-Kuhn-Tucker (KKT) conditions or primal-dual (PD) formulation which leads to introduce new binary variables or bilinear terms. In doing so, a bilevel model is obtained which cannot be solved by KKT or PD due to the presence of binary variables and thus it is required to be solved using decomposition-based methods such as Benders decomposition or column-and-constraint generation algorithm. This paper aims at proposing a solution technique to deal with the robust transmission expansion planning problem where the lower level problem is replaced by its dual problem. The advantage of dualizing the lower level problem is that all the uncertain parameters are integrated into the objective function which can be further recast as a linear constraint. Afterwards, the polyhedron uncertainty set is handled using duality theory. Since neither binary variables nor bilinear terms are introduced in this approach, the lower level can be substituted using PD formulation rendering a single level mixed-integer linear programming problem. Numerical experiments on two test systems verify that the proposed model is superior compared with the existing techniques in terms of computational burden.

Nash equilibria in electricity pool markets with large-scale wind power integration

This work investigates the interaction between power producers with conventional and wind generation portfolios participating in a network-constrained pool-based market. A stochastic bi-level problem is introduced to model the strategic behavior of each single producer. The upper-level problem maximizes the producers’ expected profits and the lower-level problem optimizes the jointly cleared energy and balancing market under economic dispatch. Market participants’ offers are modeled using linear stepwise curves, and the stochastic wind power generation is realized through a set of plausible wind scenarios. The bi-level problem is recast into a single-level mathematical problem with equilibrium constraints with primal-dual formulation using the Karush-Kuhn-Tacker first order optimality conditions and the strong duality theorem. The joint solution of all strategic producers’ problems constitutes an equilibrium problem with equilibrium constraints. The latter is reduced into an equivalent mixed integer linear program by using disjunctive constraints. Different objective functions are applied to the final program to define the range of market equilibria, and a single-iterate diagonalization process is used to justify those equilibria that are meaningful. The model addresses several cases considering different types of market competition, transmission line congestions, and different levels of wind power penetration and volatility.

A coordinated scheme for transmission and distribution expansion planning: A Tri-level approach

In traditional power system planning problems, transmission and distribution networks are separated, which is not suitable for future power systems due to the rapid development of active distribution networks. This paper presents a hierarchy framework to optimize the transmission and distribution network expansion in a simultaneous manner. The proposed decentralized decision-making architecture forms a Tri-level structure where the first level is the transmission expansion planning (TEP) problem solved by the transmission system operator (TSO) as the leader, while the second level is the distribution expansion planning (DEP) problem solved by the distribution system operators (DSOs) as the second level followers. The third level problem is the optimal economic dispatch problem solved by the independent system operator (ISO) as the third level follower. The economic dispatch problem is replaced by its primal-dual (PD) formulation leading to a bilevel multi-follower mixed-integer linear programming (BMF-MILP) problem. Then, by using multi-parametric programming (MPP) method, the BMF-MILP problem is recast as several single level independent optimization problems. Numerical experiments are shown to corroborate the efficiency and tractability of the presented model. The results show the new coordinated transmission and distribution expansion planning (CTDEP) model provides additional flexibility and reduces the total cost.

Coordinated scheme for expansion planning of distribution networks: a bilevel game approach

The expansion planning of a distribution network (DN) is studied separately and independently from the behaviour of the transmission network (TN) as well as other DNs. However, DNs are being converted from passive to active mode by extensively integrating distributed generations (DGs) into them. Hence, an uncoordinated expansion planning may lead to redundant investment cost and suboptimal solutions. Accordingly, the main motivation of this study is to propose a coordinated expansion scheme among distribution companies (DISCOs). In doing so, a novel bilevel game-based framework is presented to coordinate the decisions of DISCOs for distribution expansion planning problem. Primal–dual formulation is used to convert the bilevel problem into a single level one resulting in a discretely-constrained generalised Nash equilibrium problem. The generalised Nash equilibrium problem is further formulated as a mixed-integer linear program using Karush–Kuhn–Tucker conditions. The methodology proposed is applied to the IEEE 24-bus RTS with several 24-bus DNs. The results show that ignoring the action of other DISCOs may lead to the over- or under-investment in DG sector which consequently decreases the DISCOs’ profit. The results also show that congestion in transmission lines can considerably affect the strategic decisions of DISCOs on DG investment.

Distribution expansion planning in the presence of plug-in electric vehicle: A bilevel optimization approach

Popularization of plug-in electric vehicles has changed the traditional strategies for distribution expansion planning problem. Therefore, considering the high penetration of electric vehicles, it is necessary to propose new models and strategies for this problem. The present study proposes a new multi-stage model to expand the distribution network based on the bilevel optimization taking the plug-in electric vehicles into account. In the upper level, whose objective function is the total annualized investment cost plus annual expected production and maintenance cost, the investment and utilization decision variables are identified. In comparison, in the lower problem, whose objective function is the cost of purchased energy from the main grid, the daily optimal scheduling is determined. By using primal-dual formulation, this bilevel optimization is recast as a mixed-integer linear programming problem, which can be solved by off-the-shelf solvers. A 24-node distribution network is used to show the efficiency, tractability, and economic advantages of the proposed methodology. The results indicate that the DEP with the smart charging strategy will lead to less investment cost compared with the case in which the dumb charging strategy is utilized. In addition, the results show that the optimal solution of the proposed bilevel model is superior to the optimal solution of the model in which the smart charging problem is solved in a separate optimization problem.

A Primal-dual Learning Algorithm for Personalized Dynamic Pricing with an Inventory Constraint

We consider the problem of a firm seeking to use personalized pricing to sell an exogenously given stock of a product over a finite selling horizon to different consumer types. We assume that the type of an arriving consumer can be observed but the demand function associated with each type is initially unknown. The firm sets personalized prices dynamically for each type and attempts to maximize the revenue over the season. We provide a learning algorithm that is near-optimal when the demand and capacity scale in proportion. The algorithm utilizes the primal-dual formulation of the problem and learns the dual optimal solution explicitly. It allows the algorithm to overcome the curse of dimensionality (the rate of regret is independent of the number of types) and sheds light on novel algorithmic designs for learning problems with resource constraints.

The adaptive algorithm for the selection of sources of the method of fundamental solutions

Despite all the efforts and success for finding the optimal location of the sources outside the domain for the method of fundamental solutions (MFS), this issue continues to attract the attention from researchers for seeking more efficient and reliable algorithms. In this paper, we propose to extend the adaptive greedy technique which applies the primal-dual formulation for the selection of source nodes in the MFS for Laplace equation with nonharmonic boundary conditions. Such approach is a data-dependent algorithm which adaptively selects the suitable source nodes based on the specific adaptive procedure. Both 2D and 3D examples are provided. Moreover, the proposed algorithm is easy to implement with high accuracy.