arXiv subjects

Burak Kocuk

Publications and source records attributed to Burak Kocuk.

At least 19 recordsLinked to original sources

arXiv2026

A variational quantum eigensolver-based cutting plane framework for semidefinite programming problems

Semidefinite programming plays a key role in optimization, with broad impact across control theory, machine learning, and combinatorial optimization. Although semidefinite programs are polynomially solvable, several commonly used algorithms rest on a linear-algebraic step whose running time grows cubically with the matrix dimension and which requires the matrix itself to be held in memory, at quadratic cost. In this study, we propose replacing it with a variational quantum eigensolver, whose qubit requirement is logarithmic in the matrix dimension, and present the first end-to-end implementation of such an approach within a cutting-plane framework, together with an operator-derived ansatz whose entanglement structure is read directly from the Pauli support of the candidate matrix. Evaluated on the control family of SDPLIB against an identical scheme driven by an exact eigendecomposition, the variational oracle produces valid cuts throughout, closing 32 to 82% of the initial optimality gap against a near-constant 75 to 82% for the exact oracle. Implementing and measuring the method end to end surfaces several effects not visible from theoretical analyses alone: where memory is actually consumed, how the padding required to fit a matrix onto a quantum register can mislead the variational optimizer, and why the candidate matrices prove dense in the Pauli basis, reducing the operator-derived ansatz to full entanglement. We report these findings and discuss their implications for near-term hybrid quantum-classical approaches.

quant-ph↗

arXiv2026

Benders Decomposition with Partial Non-Anticipativity Relaxation for Multi-Stage Stochastic Clean Energy Transition Planning

We study clean energy transition planning for campus-scale integrated electricity-heat systems under both strategic level and operational level uncertainties. We formulate a multi-stage stochastic mixed-integer program that jointly optimizes investment and operational decisions for renewable generation, storage, and heat-transfer technologies whose costs and efficiencies evolve stochastically across stages. To account for short-term operational uncertainty, we further derive a robust reformulation based on box uncertainty sets for demand, renewable generation, and heat-transfer performance. To solve the resulting large-scale model, we develop a Benders decomposition algorithm with partial non-anticipativity relaxation. Investment variables and their non-anticipativity constraints are retained in the master problem while operational variables are assigned to scenario-wise subproblems where their non-anticipativity constraints are relaxed. Non-anticipativity of operational variables is restored only at termination through a smaller linear program. We prove that this correction step can only increase the objective function value by a finite bound. We improve the computational performance of the algorithm developed with valid inequalities and a two-phase cut-addition strategy. Using the proposed approach, we solve the Middle East Technical University campus case study at high temporal resolution within a reasonable computational budget, which is not otherwise possible with the extensive form or the classical Benders decomposition. We evaluate the resulting investment plans through rolling-horizon Monte Carlo simulations with an out-of-sample analysis and demonstrate that the added robustness can significantly improve the operational reliability of the transition plans with a moderate increase in total cost.

math.OC↗

arXiv2026

The Weighted Connected p-Median Problem

The connected p-median problem is defined as a variant of the classical p-median problem when the facility nodes induce a connected subgraph. In this paper, we introduce the weighted version of the above problem when the weight of the facility connection in the objective function is defined by the minimum weight spanning tree of the facility nodes. This approach is motivated by the sink node selection in distributed sensor networks, in which the collected information is shared among the sink nodes through the minimum spanning tree. The weights of the graph determining the network topology of the candidate sink nodes as connection costs are distinguished from the standard access costs of the p-median problem. The fixed deployment costs for the setup of facilities are also considered. The objective is to minimize the overall cost as the sum of deployment cost, access cost and connection cost. We show that the problem is NP-hard and propose three mixed-integer linear programming (MILP) formulations adapted from the traveling salesperson problem literature. Since these formulations are poorly scalable with respect to network size, we develop a four-phase matheuristic method based on linear programming rounding. We conduct an extensive computational study to evaluate the performance of the MILP formulations and 22 variants of the matheuristic under different parameter settings. The results indicate that the MILP models perform effectively on small instances but struggle to solve medium- and large-scale instances within a two-hour time limit. In contrast, several matheuristic variants consistently produce high-quality solutions within minutes. Finally, we analyze the impact of network structure, size, density, and the parameter $p$ on solution quality, providing further insights for network design.

math.OC↗

arXiv2026

Copositive Matrices with Ordered Off-Diagonal Entries

We study copositive matrices which admit a decomposition into a sum of a positive semidefinite matrix and a matrix with nonnegative entries. Our main result shows that if the off-diagonal entries of a copositive matrix are nondecreasing in rows and in columns, then it admits such a decomposition. We apply this result to study optimization of quadratic forms over the standard simplex. As a corollary, we obtain that a natural relaxation of this problem is tight when the objective function is separable, resolving an open question of Dey and Kocuk.

math.OC↗

arXiv2026

MIQCP and MISOCP-Based Solution Methods for the Multi-Layer Thin Films Problem

The Multi-Layer Thin Films Problem is a materials science problem that aims to enhance the reflectance of a metallic substrate by designing multi-layer coatings composed of different dielectric materials and thicknesses. While previous studies on the problem mostly rely on heuristic approaches and are designed for single wavelength applications, this work addresses the problem using global optimization techniques for multiple wavelengths. We develop an exact nonconvex mixed-integer quadratically constrained programming (MIQCP) model to solve this problem. We also develop a mixed-integer second-order cone programming relaxation that has computational advantage over the MIQCP model. Our numerical experiments yield solutions that have average reflectance of 99% over the visible spectrum (380-770 nm) and 95% over the broad spectrum (300-3000 nm).

math.OC↗

arXiv2025

Solving the Heilbronn Triangle Problem using Global Optimization Methods

We study the Heilbronn triangle problem, which involves placing n points in the unit square such that the minimum area of any triangle formed by these points is maximized. A straightforward maximin formulation of this problem is highly non-linear and non-convex due to the existence of bilinear terms and absolute value equations. We propose two mixed-integer quadratically constrained programming (MIQCP) and one QCP formulation, which can be readily solved by any global optimization solver. We develop several formulation enhancements in the form of bound tightening and symmetry breaking inequalities that are prevalent in the global optimization literature in addition to other enhancements that exploit the problem structure. With the help of these enhancements, our models reproduce proven optimal values for instances up to n = 8 points with certified optimality in the order of seconds. In the case of n = 9 points, for which no analytical proof is known, we establish a certified optimal value by a computational effort of one day. This is a significant improvement over the previous benchmark established in 31 days of computations by Chen et al. (2017).

cs.CG↗

arXiv2025

Convexification of a Separable Function over a Polyhedral Ground Set

In this paper, we study the set $\mathcal{S}^\kappa = \{ (x,y)\in\mathcal{G}\times\mathbb{R}^n : y_j = x_j^\kappa , j=1,\dots,n\}$, where $\kappa > 1$ and the ground set $\mathcal{G}$ is a nonempty polytope contained in $[0,1]^n$. This nonconvex set is closely related to separable standard quadratic programming and appears as a substructure in potential-based network flow problems from gas and water networks. Our aim is to obtain the convex hull of $\mathcal{S}^\kappa$ or its tight outer-approximation for the special case when the ground set $\mathcal{G}$ is the standard simplex. We propose power cone, second-order cone and semidefinite programming relaxations for this purpose, which are further strengthened by the Reformulation-Linearization Technique and the Reformulation-Perspectification Technique. For $\kappa=2$, we obtain the convex hull of $\mathcal{S}^\kappa$ in the low-dimensional setting. For general $\kappa$, we give approximation guarantees for the power cone representable relaxation, the weakest relaxation we consider. We prove that this weakest relaxation is tight with probability one as $n\to\infty$ when a uniformly generated linear objective is optimized over it. Finally, we provide the results of our extensive computational experiments comparing the empirical strength of several conic programming relaxations that we propose.

math.OC↗

arXiv2025

A Stochastic Programming Approach to the Railcar Maintenance Problem with Service Level and Track Capacity Considerations

Railcars, as part of the rolling stock, perform regular transportation tasks with respect to a service level agreement (SLA) and undergo preventive maintenance at regular intervals based on the recommendations of train manufacturers. When unexpected failures occur, they need to enter corrective maintenance immediately. However, this reactive approach may result in large SLA violations and an excessive number of corrective maintenance actions. In this study, we utilize a predictive maintenance approach based on the reliability of a railcar. In particular, we propose a stochastic programming model, in which railcar failure scenarios are generated from a Weibull distribution, a common assumption in the reliability literature. The model incorporates both SLA and track-capacity considerations and is solved through the Sample Average Approximation (SAA) method. We generate random instances to compare the stochastic model and a deterministic model adopted from the literature with respect to several system parameters. Our results show that the stochastic model achieves lower total costs, fewer SLA violations, and a reduced number of corrective interventions compared with deterministic approaches, while effectively managing track-capacity constraints. Our results underscore the importance of the predictive approach in the context of the railcar maintenance problem.

math.OC↗

arXiv2025

A Dynamic Strategic Plan for Transition to Campus-Scale Clean Electricity Using Multi-Stage Stochastic Programming

The decarbonization of energy systems at energy-intensive sites is an essential component of global climate mitigation, yet such transitions involve substantial capital requirements, ongoing technological progress, and the operational complexities of renewable integration. This study presents a dynamic strategic planning framework that applies multi-stage stochastic programming to guide clean electricity transitions at the campus level. The model jointly addresses technology investment, storage operation, and grid interaction decisions while explicitly incorporating uncertainties in future technology cost trajectories and efficiency improvements. By enabling adaptive, stage-wise decision-making, the framework provides a structured approach for large electricity consumers seeking to achieve self-sufficient and sustainable energy systems. The approach is demonstrated through a case study of Middle East Technical University (Ankara, Turkey), which has committed to achieving carbon-neutral electricity by 2040. Through the integration of solar photovoltaics, wind power, and lithium-ion batteries, the model links long-term investment planning with operational-level dynamics by incorporating high-resolution demand and meteorological data. Our findings from the case study, sensitivity analyses, and comparisons with simplified models indicate that accounting for uncertainty and temporal detail is crucial for both the economic viability and operational feasibility of campus-scale clean electricity transitions.

math.OC↗

arXiv2025

Global Optimization of Gas Transportation and Storage: Convex Hull Characterizations and Relaxations

Gas transportation and storage has become one of the most relevant and important optimization problems in energy systems. This problem inherently includes highly nonlinear and nonconvex aspects due to gas physics, and discrete aspects due to the control decisions of active network elements. Obtaining even locally optimal solutions for this problem presents significant mathematical and computational challenges for system operators. In this paper, we formulate the gas transportation and storage problem as a nonconvex mixed-integer nonlinear program (MINLP) through disjunctions on the flow directions. Moreover, we study the nonconvex sets induced by gas physics and propose mixed-integer second-order cone programming relaxations for the nonconvex MINLP problem. The proposed relaxations are based on the convex hull representations of two nonconvex sets: Firstly, we give the convex hull representation of the nonconvex set for pipes and show that it is second-order cone representable. Secondly, we also give a complete characterization of the extreme points of the nonconvex set for compressors and show that the convex hull of the extreme points is power cone representable. Moreover, for practical applications, we propose a second-order cone outer-approximation for the nonconvex set for compressors. To obtain (near) globally optimal solutions, we develop an algorithmic framework based on our convex hull results. We evaluate our framework through extensive computational experiments on various GasLib networks in comparison with the convex relaxations from the literature and a state-of-the-art global solver. Our results highlight the computational efficiency and convergence performance of our convex relaxation method compared to other methods. Moreover, our method also consistently provides (near) global solutions as well as high-quality warm-starting points for local solvers.

math.OC↗

arXiv2025

Risk-Averse Antibiotics Time Machine Problem

Antibiotic resistance, which is a serious healthcare issue, emerges due to uncontrolled and repeated antibiotic use that causes bacteria to mutate and develop resistance to antibiotics. The Antibiotics Time Machine Problem aims to come up with treatment plans that maximize the probability of reversing these mutations. Motivated by the severity of the problem, we develop a risk-averse approach and formulate a scenario-based mixed-integer linear program with a conditional value-at-risk objective function. We propose a risk-averse scenario batch decomposition algorithm that partitions the scenarios into manageable risk-averse subproblems, enabling the construction of lower and upper bounds. We develop several algorithmic enhancements in the form of stronger no-good cuts and symmetry breaking constraints in addition to scenario regrouping and warm starting. We conduct extensive computational experiments for static and dynamic versions of the problem on a real dataset and demonstrate the effectiveness of our approach. Our results suggest that risk-averse solutions can achieve significantly better worst-case performance compared to risk-neutral solutions with a slight decrease in terms of the average performance, especially for the dynamic version. Although our methodology is presented in the context of the Antibiotics Time Machine Problem, it can be adapted to other risk-averse problem settings in which the decision variables come from special ordered sets of type one.

math.OC↗

arXiv2025

Learning to Relax Nonconvex Quadratically Constrained Quadratic Programs

Quadratically constrained quadratic programs (QCQPs) are ubiquitous in optimization: Such problems arise in applications from operations research, power systems, signal processing, chemical engineering, and portfolio theory, among others. Despite their flexibility in modeling real-life situations and the recent effort to understand their properties, nonconvex QCQPs are hard to solve in practice. Most of the approaches in the literature are based on either Linear Programming (LP) or Semidefinite Programming (SDP) relaxations, each of which works very well for some problem subclasses but perform poorly on others. In this paper, we develop a relaxation selection procedure for nonconvex QCQPs that can adaptively decide whether an LP- or SDP-based approach is expected to be more beneficial by considering the instance structure. The proposed methodology relies on utilizing machine learning methods that involve features derived from spectral properties and sparsity patterns of data matrices, and once trained appropriately, the prediction model applies to any instance with an arbitrary number of variables and constraints. We develop classification and regression models under different feature-design setups, including a dimension-independent representation, and evaluate them on both synthetically generated instances and benchmark instances from MINLPLib. Our computational results demonstrate the effectiveness of the proposed approach for predicting the more favorable relaxation across diverse QCQP families.

math.OC↗

arXiv2024

On the integrality gap of convex mixed-integer programs

We study the integrality gap of convex mixed-integer programs, that is, the difference between the optimal value of such a problem and the optimal value of its continuous relaxation. We study classes of convex sets whose associated optimization problem have finite integrality gap: Dirichlet convex sets, sets with full-dimensional recession cones and sets that can be approximated by polyhedral sets. In the latter two cases, we provide estimates for the value of the integrality gap. Finally, we study the possibility of estimating the integrality gap of nonlinear convex mixed-integer programs via rational polyhedral approximations of their feasible regions and argue that, in general, such an approach may yield arbitrarily worse bounds compared to integrality gap estimations specifically derived by studying the associated nonlinear set.

math.OC↗

arXiv2024

A Dynamic Strategic Plan for the Transition to a Clean Bus Fleet using Multi-Stage Stochastic Programming with a Case Study in Istanbul

In recent years, the transition to clean bus fleets has accelerated. Although this transition might bring environmental and economic benefits, it requires a long-term strategic plan due to the large investment costs involved. This paper proposes a multi-stage stochastic program to optimize strategic plans for the clean bus fleet transition that explicitly considers the uncertainty scenarios in the cost and efficiency improvements of clean buses. Our optimization model minimizes the total expected cost subject to emission targets, budget restrictions and several other operational considerations. We propose a new forecasting approach that captures the correlation between these improvements to obtain realistic future pathways for Battery Electric Buses (BEBs) and Hydrogen Fuel Cell Buses (HFCBs), which are then given to the multi-stage stochastic program as scenarios. We also utilize a physics-based model for BEBs to accurately capture their energy consumption and recharging needs. As a case study, we focus on the complex public bus network of Istanbul, which aims to transition to a clean bus fleet by 2050. Utilizing real datasets, we solve a five-stage stochastic program spanning a 25-year planning horizon that involves 256 scenarios to obtain dynamic strategic plans that can be used by the policy makers. Our results suggest that BEBs are more advantageous than HFCBs, even in slow BEB but fast HFCB development scenarios. We also conduct several sensitivity analyses to understand the effects of the intermediate emission targets, budget limitations and energy prices.

math.OC↗

arXiv2024

Generator Subadditive Functions for Mixed-Integer Programs

For equality-constrained linear mixed-integer programs (MIP) defined by rational data, it is known that the subadditive dual is a strong dual and that there exists an optimal solution of a particular form, termed generator subadditive function. Motivated by these results, we explore the connection between Lagrangian duality, subadditive duality and generator subadditive functions for general equality-constrained MIPs where the vector of variables is constrained to be in a monoid. We show that strong duality holds via generator subadditive functions under certain conditions. For the case when the monoid is defined by the set of all mixed-integer points contained in a convex cone, we show that strong duality holds under milder conditions and over a more restrictive set of dual functions. Finally, we provide some examples of applications of our results.

math.OC↗

arXiv2023

Discretization-Based Solution Approaches for the Circle Packing Problem

The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing problem can be cast as a nonconvex quadratically constrained program, and is difficult to solve in general. An iterative solution approach based on a bisection-type algorithm on the radius of the larger circle is provided. The present algorithm discretizes the container into small cells and solves two different integer linear programming formulations proposed for a restricted and a relaxed version of the original problem. The present algorithm is enhanced with solution space reduction, bound tightening and variable elimination techniques. Then, a computational study is performed to evaluate the performance of the algorithm. The present algorithm is compared with BARON and Gurobi that solve the original nonlinear formulation and heuristic methods from literature, and obtain promising results.

math.OC↗

arXiv2023

Improved Rank-One-Based Relaxations and Bound Tightening Techniques for the Pooling Problem

The pooling problem is a classical NP-hard problem in the chemical process and petroleum industries. This problem is modeled as a nonlinear, nonconvex network flow problem in which raw materials with different specifications are blended in some intermediate tanks, and mixed again to obtain the final products with desired specifications. The analysis of the pooling problem is quite an active research area, and different exact formulations, relaxations and restrictions are proposed. In this paper, we focus on a recently proposed rank-one-based formulation of the pooling problem. In particular, we study a recurring substructure in this formulation defined by the set of nonnegative, rank-one matrices with bounded row sums, column sums, and the overall sum. We show that the convex hull of this set is second-order cone representable. In addition, we propose an improved compact-size polyhedral outer-approximation and families of valid inequalities for this set. We further strengthen these convexification approaches with the help of various bound tightening techniques specialized to the instances of the pooling problem. Our computational experiments show that the newly proposed polyhedral outer-approximation can improve upon the traditional linear programming relaxations of the pooling problem in terms of the dual bound. Furthermore, bound tightening techniques reduce the computational time spent on both the exact, linear programming and mixed-integer linear programming relaxations.

math.OC↗

arXiv2022

A Joint Chance-Constrained Stochastic Programming Approach for the Integrated Predictive Maintenance and Operations Scheduling Problem in Power Systems

Maintenance planning plays a key role in power system operations under uncertainty by helping system operators ensure a reliable and secure power grid. This paper studies a short-term condition-based integrated maintenance planning with operations scheduling problem while considering the unexpected failure possibilities of generators as well as transmission lines. We formulate this problem as a two-stage stochastic mixed-integer program with failure scenarios sampled from the sensor-driven remaining lifetime distributions of the individual system elements whereas a joint chance-constraint consisting of Poisson Binomial random variables is introduced to account for failure risks. Because of its intractability, we develop a cutting-plane method to obtain an exact reformulation of the joint chance-constraint by proposing a separation subroutine and deriving stronger cuts as part of this procedure. To solve large-scale instances, we derive a second-order cone programming based safe approximation of this constraint. Furthermore, we propose a decomposition-based algorithm implemented in parallel fashion for solving the resulting stochastic program, by exploiting the features of the integer L-shaped method and the special structure of the maintenance and operations scheduling problem to derive stronger optimality cuts. We further present preprocessing steps over transmission line flow constraints to identify redundancies. To illustrate the computational performance and efficiency of our algorithm compared to more conventional maintenance approaches, we design a computational study focusing on a weekly plan with daily maintenance and hourly operational decisions involving detailed unit commitment subproblems. Our computational results on various IEEE instances demonstrate the computational efficiency of the proposed approach with reliable and cost-effective maintenance and operational schedules.

math.OC↗