Mikhail A. Bragin

Convergence of the Surrogate Lagrangian Relaxation Method

Studies have shown that the surrogate subgradient method, to optimize non-smooth dual functions within the Lagrangian relaxation framework, can lead to significant computational improvements as compared to the subgradient method. The key idea is to obtain surrogate subgradient directions that form acute angles toward the optimal multipliers without fully minimizing the relaxed problem. The major difficulty of the method is its convergence, since the convergence proof and the practical implementation require the knowledge of the optimal dual value. Adaptive estimations of the optimal dual value may lead to divergence and the loss of the lower bound property for surrogate dual values. The main contribution of this paper is on the development of the surrogate Lagrangian relaxation method and its convergence proof to the optimal multipliers, without the knowledge of the optimal dual value and without fully optimizing the relaxed problem. Moreover, for practical implementations, a stepsizing formula that guarantees convergence without requiring the optimal dual value has been constructively developed. The key idea is to select stepsizes in a way that distances between Lagrange multipliers at consecutive iterations decrease, and as a result, Lagrange multipliers converge to a unique limit. At the same time, stepsizes are kept sufficiently large so that the algorithm does not terminate prematurely. At convergence, the lower-bound property of the surrogate dual is guaranteed. Testing results demonstrate that non-smooth dual functions can be efficiently optimized, and the new method leads to faster convergence as compared to other methods available for optimizing non-smooth dual functions, namely, the simple subgradient method, the subgradient-level method, and the incremental subgradient method.

Payment cost minimization using Lagrangian relaxation and modified surrogate optimization approach

In the Payment Cost Minimization (PCM) mechanism [4] payment costs are minimized directly, thus the payment costs that results from selected offers can be significantly reduced compared to the costs obtained by minimizing total bid costs. The PCM can be solved efficiently using standard LP software packages (e.g., CPLEX) only for a limited number of offers. Lagrangian relaxation (LR) has been a powerful technique to solve discrete and mixed-integer optimization problems. For complex problems, such as the PCM, the surrogate subgradient method is frequently used within Lagrangian relaxation approach to update multipliers (e.g., [6], [4]). In the surrogate subgradient approach a proper direction is obtained without fully minimizing the relaxed problem. This paper presents a modified Lagrangian relaxation and the surrogate optimization approach for obtaining a good feasible solution within a reasonable CPU time. The difficulty of the standard surrogate optimization method primarily arises due to the lack of prior knowledge about the optimal dual value, which is used in the definition of a step size. In order to overcome this difficulty, a new method is proposed. The main purpose of the modified surrogate subgradient approach is to obtain a “good” direction quickly and independently of the optimal dual value at each iteration. In this paper it is achieved by introducing a formula for updating the multipliers such that the exact minimization of the Lagrangian leads to a convergent result. Then an approximate formula for updating the multipliers is developed so that the exact optimization of the Lagrangian leads to a convergent result under certain optimality conditions. Lastly, the notion of the surrogate subgradient is used for ensuring the convergence within the reasonable CPU time. An analogue of the surrogate subgradient condition guarantees the convergence on the surrogate subgradient method. Numerical examples are provided to demonstrate the methods effectiveness.

An efficient surrogate subgradient method within Lagrangian relaxation for the Payment Cost Minimization problem

Studies have shown that for a given set of bids, Payment Cost Minimization leads to lower customer payments as compared to Bid Cost Minimization. In order to provide a thorough analysis of the two mechanisms an efficient solution methodology is required. It has previously been shown that the surrogate optimization within the Lagrangian relaxation framework can lead to savings in the CPU time while ensuring a high-quality solution. This paper develops an efficient methodology to solve Payment Cost Minimization using the surrogate optimization framework and the branch-and-cut method. In the presented methodology the problem structure is exploited using Lagrangian relaxation and the relaxation of the integrality constraints is exploited using branch-and-cut. The resulting method is further improved by using additional cutting planes that reduce the search space and by the advanced start to reinitialize the decision variables at each iteration. For large Payment Cost Minimization problems, the method can find significantly better feasible solutions within less CPU time than that obtained by standard branch-and-cut methods implemented in commercial MIP solver. The methodology developed in this paper is generic and can be used for solving other optimization problems.

An efficient approach for solving mixed-integer programming problems under the monotonic condition

Many important integer and mixed-integer programming problems are difficult to solve. A representative example is unit commitment with combined cycle units and transmission capacity constraints. Complicated transitions within combined cycle units are difficult to follow, and system-wide coupling transmission capacity constraints are difficult to handle. Another example is the quadratic assignment problem. The presence of cross-products in the objective function leads to nonlinearity. In this study, building upon the novel integration of surrogate Lagrangian relaxation and branch-and-cut, such problems will be solved by relaxing selected coupling constraints. Monotonicity of the relaxed problem will be assumed and exploited and nonlinear terms will be dynamically linearised. The linearity of the resulting problem will be exploited using branch-and-cut. To achieve fast convergence, guidelines for selecting stepsizing parameters will be developed. The method opens up directions for solving nonlinear mixed-in...

Novel exploitation of convex hull invariance for solving unit commitment by using surrogate Lagrangian relaxation and branch-and-cut

Many important problems in power systems, including unit commitment and economic dispatch (UCED), are modeled as MILP problems which are computationally intensive. Such problems can often be viewed as subsystems coupled by system-wide constraints. These structures have been efficiently exploited by surrogate Lagrangian relaxation and branch-and-cut to solve UCED problems with combined cycle units by relaxing system-wide coupling constraints, decomposing the relaxed problem into subproblems, and solving each subproblem by branch-and-cut. However, while complicated features of a subproblem are handled locally within that subproblem, there is no guarantee that each subproblem can be efficiently solved because corresponding convex hulls are generally difficult to obtain, and the overall computational effort may be significant. In this paper, this difficulty is alleviated by the novel exploitation of the invariance of subproblem constraints and the associated convex hulls with respect to updating of Lagrange multipliers. Consequently, cuts remain valid throughout the entire iterative process. When cuts are retained, solving subproblems in subsequent iterations becomes easier than starting from scratch. This idea is operationalized in CPLEX by using C Concert Technology to extract, save and load CPLEX-generated cuts by using callable libraries. Numerical results demonstrate that the new approach is computationally efficient and generates good feasible solutions.

Surrogate Lagrangian relaxation and branch-and-cut for unit commitment with combined cycle units

Combined cycle (CC) units are efficient because heat from combustion turbines is not wasted but is used for steam turbines. However, when state transitions are followed, CC units complicate the unit commitment and economic dispatch (UCED) problem. While branch-and-cut has been successful in solving UCED problems without considering CC units, transitions in one such unit affect the entire problem. Therefore, the convex hull is difficult to obtain and the UCED problem with CC units is difficult to solve. To efficiently solve the problem, we exploit linearity as well as separability. To decompose the problem into subproblems associated with conventional and CC units, our recently developed surrogate Lagrangian relaxation will be used to relax coupling system-wide constraints, and each subproblem will then be solved by using branch-and-cut. Constraints as well as transitions within a CC unit are handled locally and no longer affect the entire problem. Moreover, we will demonstrate that branch-and-cut can handle individual subproblems much more efficiently as compared to the original problem. The linear structure of coupling constraints is then exploited to obtain feasible costs. Numerical results demonstrate that the new approach is computationally efficient and generates good feasible solutions.

Energy-efficient building clusters

With worlds increasing energy demand and growing environmental concerns, efficient utilization of energy is critical for sustainable living. Buildings are the major energy consumers, and a set of buildings are connected by distributed energy resources (DERs) such as chillers and boilers in certain building clusters, e.g., a school campus or residential community. Optimized operation of such building clusters, however, is challenging. Proper device and building models are needed, and it is difficult and impractical to control everything at the cluster level. This paper presents integrated optimization of building clusters with buildings and DERs to reduce energy costs and CO2 emissions. From the energy and emission point of view, energy networks of buildings, devices, and water/electricity networks are established for energy generation, conversation, storage and utilization, and emissions. The problem is to match different types of energy demand of buildings and supplies from devices with time-varying electricity and gas prices. A mixed-integer model for a small building cluster is established. To coordinate buildings and devices, our idea is to use multipliers as shadow prices in a decomposition and coordination structure. Our surrogate Lagrangian relaxation method is used to solve the problem. With breakthroughs in multiplier updating directions and step-sizing formulas, computational efforts are much reduced. Preliminary optimization and simulation results show that total energy costs and emissions are reduced by optimized operation, e.g., making direct use of energy resource and avoiding unnecessary energy conversion.

Efficient surrogate optimization for payment cost co-optimization with transmission capacity constraints

Most ISOs in the US minimize the total bid cost and then settle the market based on locational marginal prices. Minimizing the total bid cost, however, may not lead to maximizing the social welfare. Studies indicate that for energy only, payment cost minimization (PCM) leads to reduced payments for a given set of bids, and the “hockey-stick” bidding a less likely to occur. Since co-optimization of energy with ancillary services leads to a more efficient allocation, it is important to solve PCM co-optimization while considering transmission capacity constraints for a comparison with other auction mechanisms. In this paper, PCM is formulated using price definition system-wide constraints. This problem formulation introduces difficulties such as nonlinearity, non-separability and complexity of convex hull. To overcome these difficulties, the nonlinear terms are linearized thereby allowing to be solved by using branch-and-cut. At the same time, the linearization is performed in a way that the “surrogate optimality condition” is satisfied thereby allowing the problem to be solved efficiently.

Exergy-efficient management of energy districts

Sustainable development requires not only the use of sustainable energy resources, but also the efficient use of all energy resources. The latter should be reached by considering the concept of energy as well as exergy - the true magnitude of thermodynamic losses. Exergy describes the quality of an energy flow as the percentage that can be completely transformed into any other form of energy. Reduction of exergy losses represents a more efficient use of energy resources, which is essential in the long run, but it is not captured by standard energy costs, which are crucial in the short run. In this paper, exergy analysis is used in the context of a multi-carrier energy district to match the supply and demand not only in quantity but also in quality. The innovative contribution of this paper is the offering of a trade-off between reducing exergy losses and energy costs, thereby attaining sustainability of the energy district. A mixed-integer programming problem considering several energy devices is formulated to minimize a weighted sum of exergy losses and energy costs while satisfying time-varying user demands. The problem is solved by branch-and-cut. Numerical results demonstrate that the optimized operation of the energy devices makes the energy district sustainable in terms of exergy efficiency and costs.

Solving payment cost co-optimization problems

Current U.S. electricity markets select supply bids by using a bid cost minimization (BCM) auction mechanism but then settle the payments based on locational marginal prices (LMPs). The resulting payments can be significantly higher than the minimized bid costs. An alternative payment cost minimization (PCM) mechanism aiming to minimize the total payments has been discussed. Studies on single product problems have shown that PCM leads to reduced payments, but few results have been reported for the co-optimization of energy and other products. In view that co-optimization leads to a more efficient capacity allocation than optimizing each product individually, it is important to investigate the PCM co-optimization problems, and solve them in standard MIP solvers for a fair comparison with BCM. In PCM, prices are decision variables and need to be appropriately defined. We characterized marginal price-setting units by using logical constraints and converted them to linear forms since linearity is required by the standard MIP solvers. The nonlinear cross-product in PCM objective function, however, cannot be converted to linear forms. Based on our recent results on surrogate optimization, a method is developed to deal with nonlinearity. Prices are first fixed at their values at the previous iteration to obtain linear formulation, and are then updated using price definition if the surrogate condition is satisfied. Numerical testing results of small examples and a 24-bus example demonstrate the effectiveness and efficiency of the method.