arXiv subjects

Shudian Zhao

Publications and source records attributed to Shudian Zhao.

10 recordsLinked to original sources

arXiv2026

ADMM-based decomposed DNN+RLT Relaxations for Completely Positive Models in Electricity Market Clearing

The day-ahead electricity market clearing with nonconvex order types can be formulated as a mixed-integer linear program (MILP), but its LP relaxation may provide weak bounds, and exact solutions can become computationally intractable in large-scale or extended market settings. We study a welfare-maximizing clearing model with elementary hourly orders, block orders with logical acceptance constraints, and flexible hourly orders. Starting from a compact MILP formulation, we derive an equivalent completely positive programming (CPP) reformulation via matrix lifting and propose relaxed CPP variants that further reduce the modeling burden while maintaining strong bounds. We then develop tractable doubly nonnegative (DNN) relaxations, including decomposed formulations that exploit the problem structure by using smaller positive semidefinite matrices. To further strengthen these bounds, we introduce reformulation-linearization technique (RLT) inequalities tailored to the decomposed structure. To tackle the challenge of large-scale DNNs, we design an alternating direction method of multipliers (ADMM) with adaptive penalty updates and rigorous dual lower bounds, enabling certified early termination. Computational experiments on synthetic instances show that the proposed DNN+RLT relaxations substantially tighten LP bounds, while decomposition and first-order methods significantly reduce computational effort.

eess.SY↗

arXiv2026

Hybrid Quantum-Classical Optimization Workflows for the Shipment Selection Problem

We present a quantum optimization framework for the Shipment Selection Problem (SSP) in electric freight logistics, developed jointly by IonQ and Einride. Idle gaps arising from stochastic shipment cancellations reduce fleet utilization and revenue; filling them optimally requires solving a combinatorial assignment problem with quadratic inter-gap dependencies. We formulate the SSP as a Mixed-Integer Quadratic Program, map it to an Ising cost Hamiltonian, and solve it using Iterative-QAOA, a non-variational warm-start extension of the Quantum Approximate Optimization Algorithm (QAOA) with a fixed linear-ramp parameter schedule. An end-to-end hybrid workflow integrates Einride's vehicle routing problem (VRP) solver with IonQ's quantum simulations, enabling evaluation on real, anonymized logistics data spanning up to 130 qubits. We assess solution quality through application-level performance metrics, including Shipments Delivered (SD), Schedule Compatibility Score (SCS), and Total Drive Distance (TDD). When the quantum assignment is passed to the classical solver as a warm start, the resulting hybrid workflow achieves improvements of up to 12% in SD and a reduction of up to 6% in total drive distance per shipment for specific instances, while total operational cost remains effectively unchanged. For the subset of instances within reach of current devices (20-35 qubits), the workflow is additionally executed on IonQ trapped-ion quantum hardware, where the hardware results closely reproduce the noiseless simulations. These results show that Iterative-QAOA can generate compatibility-aware assignments that become operationally valuable when embedded in a hybrid logistics optimization workflow.

quant-ph↗

arXiv2026

A Framework for Eliminating Paradoxical Orders in European Day-Ahead Electricity Markets through Mixed-Integer Linear Programming Strong Duality

The presence of integer variables in the European day-ahead electricity market renders the social welfare maximization problem non-convex and non-differentiable, making classical marginal pricing theoretically inconsistent. Existing pricing mechanisms often struggle to balance revenue adequacy with incentive compatibility, typically relying on discriminatory uplift payments or suffering from weak duality. Leveraging the Augmented Lagrangian Duality (ALD) framework, which establishes strong duality for Mixed-Integer Linear Programming (MILP), this paper proposes a novel ALD pricing mechanism. We analytically prove that this mechanism is inherently incentive-compatible, aligning centralized dispatch with individual incentives without requiring side payments. Notably, we demonstrate that the ALD price signals intrinsically eliminate Paradoxically Rejected Orders (PROs) and Paradoxically Accepted Orders (PAOs) for supply orders. For the demand side, a sufficient condition is introduced to further guarantee revenue adequacy, resulting in a transparent and financially consistent settlement system. To ensure computational tractability, we modify the Surrogate Absolute-Value Lagrangian Relaxation (SAVLR) method to efficiently compute the exact penalty coefficients and optimal Lagrangian multipliers. Numerical experiments on illustrative examples and the Nordic 12-area electricity market model confirm the superior economic properties of the ALD pricing mechanism and the tractability of the modified SAVLR algorithm.

math.OC↗

arXiv2024

A constrained optimization approach to improve robustness of neural networks

In this paper, we present a novel nonlinear programming-based approach to fine-tune pre-trained neural networks to improve robustness against adversarial attacks while maintaining high accuracy on clean data. Our method introduces adversary-correction constraints to ensure correct classification of adversarial data and minimizes changes to the model parameters. We propose an efficient cutting-plane-based algorithm to iteratively solve the large-scale nonconvex optimization problem by approximating the feasible region through polyhedral cuts and balancing between robustness and accuracy. Computational experiments on standard datasets such as MNIST and CIFAR10 demonstrate that the proposed approach significantly improves robustness, even with a very small set of adversarial data, while maintaining minimal impact on accuracy.

cs.LG↗

arXiv2024

Tight SDP relaxations for cardinality-constrained problems

We model the cardinality-constrained portfolio problem using semidefinite matrices and investigate a relaxation using semidefinite programming. Experimental results show that this relaxation generates tight lower bounds and even achieves optimality on many instances from the literature. This underlines the modeling power of semidefinite programming for mixed-integer quadratic problems.

math.OC↗

arXiv2023

Alternating mixed-integer programming and neural network training for approximating stochastic two-stage problems

The presented work addresses two-stage stochastic programs (2SPs), a broadly applicable model to capture optimization problems subject to uncertain parameters with adjustable decision variables. In case the adjustable or second-stage variables contain discrete decisions, the corresponding 2SPs are known to be NP-complete. The standard approach of forming a single-stage deterministic equivalent problem can be computationally challenging even for small instances, as the number of variables and constraints scales with the number of scenarios. To avoid forming a potentially huge MILP problem, we build upon an approach of approximating the expected value of the second-stage problem by a neural network (NN) and encoding the resulting NN into the first-stage problem. The proposed algorithm alternates between optimizing the first-stage variables and retraining the NN. We demonstrate the value of our approach with the example of computing operating points in power systems by showing that the alternating approach provides improved first-stage decisions and a tighter approximation between the expected objective and its neural network approximation.

math.OC↗

arXiv2023

Ground truth clustering is not the optimum clustering

The clustering of data is one of the most important and challenging topics in data science. The minimum sum-of-squares clustering (MSSC) problem asks to cluster the data points into $k$ clusters such that the sum of squared distances between the data points and their cluster centers (centroids) is minimized. This problem is NP-hard, but there exist exact solvers that can solve such problem to optimality for small or medium size instances. In this paper, we use a branch-and-bound solver based on semidefinite programming relaxations called SOS-SDP to compute the optimum solutions of the MSSC problem for various $k$ and for multiple datasets, with real and artificial data, for which the data provider has provided ground truth clustering. Next, we use several extrinsic and intrinsic measures to evaluate how the optimum clustering and ground truth clustering matches, and how well these clusterings perform with respect to the criteria underlying the intrinsic measures. Our calculations show that the ground truth clusterings are generally far from the optimum solution to the MSSC problem. Moreover, the intrinsic measures evaluated on the ground truth clusterings are generally significantly worse compared to the optimum clusterings. However, when the ground truth clustering is in the form of convex sets, e.g., ellipsoids, that are well separated from each other, the ground truth clustering comes very close to the optimum clustering.

math.OC↗

arXiv2023

Model-based feature selection for neural networks: A mixed-integer programming approach

In this work, we develop a novel input feature selection framework for ReLU-based deep neural networks (DNNs), which builds upon a mixed-integer optimization approach. While the method is generally applicable to various classification tasks, we focus on finding input features for image classification for clarity of presentation. The idea is to use a trained DNN, or an ensemble of trained DNNs, to identify the salient input features. The input feature selection is formulated as a sequence of mixed-integer linear programming (MILP) problems that find sets of sparse inputs that maximize the classification confidence of each category. These ''inverse'' problems are regularized by the number of inputs selected for each category and by distribution constraints. Numerical results on the well-known MNIST and FashionMNIST datasets show that the proposed input feature selection allows us to drastically reduce the size of the input to $\sim$15\% while maintaining a good classification accuracy. This allows us to design DNNs with significantly fewer connections, reducing computational effort and producing DNNs that are more robust towards adversarial attacks.

math.OC↗

arXiv2022

Partitioning through projections: strong SDP bounds for large graph partition problems

The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both positive semidefinite and nonnegative, strengthened by additional polyhedral cuts for two variations of the GPP: the $k$-equipartition and the graph bisection problem. After reducing the size of the relaxations by facial reduction, we solve them by a cutting-plane algorithm that combines an augmented Lagrangian method with Dykstra's projection algorithm. Since many components of our algorithm are general, the algorithm is suitable for solving various DNN relaxations with a large number of cutting planes. We are the first to show the power of DNN relaxations with additional cutting planes for the GPP on large benchmark instances up to 1,024 vertices. Computational results show impressive improvements in strengthened DNN bounds.

math.OC↗

arXiv2022

SDP-based bounds for graph partition via extended ADMM

We study two NP-complete graph partition problems, $k$-equipartition problems and graph partition problems with knapsack constraints (GPKC). We introduce tight SDP relaxations with nonnegativity constraints to get lower bounds, the SDP relaxations are solved by an extended alternating direction method of multipliers (ADMM). In this way, we obtain high quality lower bounds for $k$-equipartition on large instances up to $n =1000$ vertices within as few as five minutes and for GPKC problems up to $n=500$ vertices within as little as one hour. On the other hand, interior point methods fail to solve instances from $n=300$ due to memory requirements. We also design heuristics to generate upper bounds from the SDP solutions, giving us tighter upper bounds than other methods proposed in the literature with low computational expense.

math.OC↗