Cone Duality Research Articles

Bi-level distributionally robust optimization model for low-carbon planning of integrated electricity and heat systems

Interdependence among diverse energy forms within integrated electricity and heat systems (IEHSs) holds the potential to harness cooperation effects. This paper proposes a bi-level optimization model for the low-carbon planning of IEHSs, which realizes the coordination between IEHSs and their upstream networks. The precise carbon emissions of IEHSs are traced through carbon emission flows of components, power distribution networks, and district heating networks. Moreover, the distributionally robust optimization approach is employed to account for the inherent uncertainty in renewable power outputs faced by IEHS planning, by incorporating the moment information into an ambiguity set. The bi-level distributionally robust planning model of IEHSs is initially reconfigured into a second-order cone structure, employing the strong duality theory, second-order cone duality theory, and linear decision rules. Subsequently, an iterative approach is introduced to attain convergence of the reformulated bi-level model. Numerical results demonstrate the effectiveness and superiority of the proposed approach for reducing carbon emissions and economic costs of IEHSs.

Data‐driven distributionally robust supplier selection and order allocation problems considering carbon emissions

AbstractUnder the constraints of uncertainty and carbon emission policies, reasonable and flexible supplier selection and order allocation can minimize the total cost, which is of great significance to the supply chain. We study sustainable supplier selection and order allocation problems with multiple periods, multiple suppliers, and multiple products under a carbon cap‐and‐trade policy. We exploit distributionally robust versions of supplier selection and order allocation problems with an explicit distributional uncertainty set and derive computationally tractable reformulations of the distributionally robust optimization model with chance constraints by cone duality theory and conditional value‐at‐risk measure. Numerical experiments show that the proposed distributionally robust optimization models with nice computational tractability and probability guarantees are superior to the existing distributionally robust chance‐constrained optimization models in terms of cost and risk aversion. Relevant management insights on different carbon emission regulations and the application of distributionally robust optimization models are provided for policymakers and enterprises' decision‐makers based on the derived results.

A robust multi-supplier multi-period inventory model with uncertain market demand and carbon emission constraint

The multi-period inventory management determines the optimal order quantity of products at the beginning of each period, which is an important research issue in supply chain management. In this paper, we propose a robust multi-supplier multi-period inventory model with uncertain market demand and uncertain unit carbon emission in transportation. We select product quality, ordering cost, service level, and emergency capacity as evaluation criteria for each supplier, employ analytic hierarchy process (AHP) technique to comprehensively evaluate and score each supplier. According to the evaluated score, the order weight for each supplier is obtained. In our model, we develop carbon emission constraint based on the carbon cap and carbon trading mechanism. As for uncertain market demand and uncertain unit carbon emission in transportation, we construct Box+Ball and Budget uncertainty sets to characterize various model uncertainties. The robust inventory model is a semi-infinite programming model, and cannot be solved directly. We transform the semi-infinite programming model into a mixed-integer second-order cone-programming (MISOCP) one via cone duality theory. Finally, the effectiveness of the proposed optimization method is illustrated by a case study.

Distributionally Robust Optimal Dispatch of Virtual Power Plant Based on Moment of Renewable Energy Resource

Virtual power plants can effectively integrate different types of distributed energy resources, which have become a new operation mode with substantial advantages such as high flexibility, adaptability, and economy. This paper proposes a distributionally robust optimal dispatch approach for virtual power plants to determine an optimal day-ahead dispatch under uncertainties of renewable energy sources. The proposed distributionally robust approach characterizes probability distributions of renewable power output by moments. In this regard, the faults of stochastic optimization and traditional robust optimization can be overcome. Firstly, a second-order cone-based ambiguity set that incorporates the first and second moments of renewable power output is constructed, and a day-ahead two-stage distributionally robust optimization model is proposed for virtual power plants participating in day-ahead electricity markets. Then, an effective solution method based on the affine policy and second-order cone duality theory is employed to reformulate the proposed model into a deterministic mixed-integer second-order cone programming problem, which improves the computational efficiency of the model. Finally, the numerical results demonstrate that the proposed method achieves a better balance between robustness and economy. They also validate that the dispatch strategy of virtual power plants can be adjusted to reduce costs according to the moment information of renewable power output.

Distributionally Robust Co-Optimization of Energy and Reserve for Combined Distribution Networks of Power and District Heating

The combined operations of power distribution and district heating networks catalyze ongoing progressions to smart cities. This paper proposes a distributionally robust energy and reserve co-optimization approach for combined power distribution and district heating networks. In particular, smart buildings are modeled as equivalent storage devices representing cost-effective resources to enhance operational flexibility and provide additional reserves in power systems. Uncertainties pertaining to renewable energy resource power and ambient temperature are captured within an ambiguity set that comprises a variety of statistical characterizations (e.g., the expected value, the variance, and the covariance) to generate more trustworthy and less conservative solutions. Linear decision rules and the second-order cone duality are employed to enhance the computational tractability of the proposed approach by transforming the original problem into a second-order cone program. Finally, numerical results provided based on the real-world Barry Island system demonstrate the risk-reduction and economic benefits of the proposed approach.

Characterizing volume via cone duality

For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between the pseudo-effective cone of divisors and the movable cone of curves. Inspired by this result, we define and study a natural intersection-theoretic volume functional for 1-cycles over compact Kahler manifolds. In particular, for numerical equivalence classes of curves over projective varieties, it is closely related to the mobility functional studied by Lehmann.

The structural physical approximation conjecture

It was conjectured that the structural physical approximation (SPA) of an optimal entanglement witness is separable (or equivalently, that the SPA of an optimal positive map is entanglement breaking). This conjecture was disproved, first for indecomposable maps and more recently for decomposable maps. The arguments in both cases are sketched along with important related results. This review includes background material on topics including entanglement witnesses, optimality, duality of cones, decomposability, and the statement and motivation for the SPA conjecture so that it should be accessible for a broad audience.

Sufficient bigness criterion for differences of two nef classes

We prove the qualitative part of Demailly's conjecture on transcendental Morse inequalities for differences of two nef classes satisfying a numerical relative positivity condition on an arbitrary compact K\"ahler (and even more general) manifold. The result improves on an earlier one by J. Xiao whose constant $4n$ featuring in the hypothesis is now replaced by the optimal and natural $n$. Our method follows arguments by Chiose as subsequently used by Xiao up to the point where we introduce a new way of handling the estimates in a certain Monge-Amp\`ere equation. This result is needed to extend to the K\"ahler case and to transcendental classes the Boucksom-Demailly-Paun-Peternell cone duality theorem if one is to follow these authors' method and was conjectured by them.

DUALITIES AND POSITIVITY IN THE STUDY OF QUANTUM ENTANGLEMENT

We present a survey on mathematical topics relating to separable states and entanglement witnesses. The convex cone duality between separable states and entanglement witnesses is discussed and later generalized to other families of operators, leading to their characterization via multiplicative properties. The condition for an operator to be an entanglement witness is rephrased as a problem of positivity of a family of real polynomials. By solving the latter in a specific case of a three-parameter family of operators, we obtain explicit description of entanglement witnesses belonging to that family. A related problem of block positivity over real numbers is discussed. We also consider a broad family of block positivity tests and prove that they can never be sufficient, which should be useful in case of future efforts in that direction. Finally, we introduce the concept of length of a separable state and present new results concerning relationships between the length and Schmidt rank. In particular, we prove that separable states of length lower or equal to 3 have Schmidt ranks equal to their lengths. We also give an example of a state which has length 4 and Schmidt rank 3.

Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

General principle of local reflexivity in the theory of cone duality

General principle of local reflexivity in the theory of cone duality