arXiv subjects

Behçet Açıkmeşe

Publications and source records attributed to Behçet Açıkmeşe.

At least 19 recordsLinked to original sources

arXiv2026

Robust Planning with Quadratically Constrained State-Dependent Uncertainties

We address robust trajectory-planning problems with state constraints and with control/state-dependent uncertain inputs and measurements. Uncertain inputs and measurements satisfy quadratic inequalities with rank-2 indefinite multipliers; measurements depend on the uncertain-state realization and on the uncertain inputs. By means of Lagrangian duality, we derive finite nonsmooth constraints that guarantee robust satisfaction of the state constraints, yielding a formulation that can be addressed without bilevel solvers. We develop a two-step convex heuristic for the resulting nonconvex nonsmooth formulation, using a convex relaxation followed by a convex feasibility-recovery step. We test the two-step approach on a rendezvous scenario with state-dependent navigation uncertainty, and we reduce by a factor 13.8 the computation time with respect to the commercial solver Gurobi, while maintaining robust satisfaction of constraints.

math.OC↗

arXiv2026

Successive Convexification for Trajectory Optimization with Continuous-time Satisfaction of Signal Temporal Logic Specifications

This paper presents a successive convexification framework for trajectory optimization under continuous-time Signal Temporal Logic (CT-STL) specifications. The framework employs generalized mean-based robustness (GMSR), a smooth and exact parameterization of discrete-time STL, as a logical building block for constructing differentiable CT-STL constraints in optimal control. It is integrated with time-dilation for free-final-time problems, finite-dimensional control parameterization, multiple-shooting discretization of the dynamics, and a convergence-guaranteed sequential convex programming method, prox-convex, to solve the nonconvex program. The main CT-STL realization embeds temporal aggregation into augmented continuous-time dynamics. This augmentation-based construction is largely transcription-independent, can be incorporated into existing optimal-control pipelines with minimal structural changes, and enables smooth CT-STL parameterizations with accuracy controlled by a user-selected regularization parameter. We also discuss a complementary dense-time realization that evaluates CT-STL formulas directly on the integration subnodes used for dynamics discretization, yielding a smooth and exact parameterization on the numerical trajectory representation, up to the accuracy of the integration scheme. The proposed GMSR-based formulations mitigate the locality and gradient-masking behavior of standard quantitative semantics and therefore provide a favorable landscape for gradient-based trajectory optimization. The framework is demonstrated through trajectory-optimization examples for a double-integrator system with continuous-time \always{}, \eventually{}, and \until{} specifications, and a 6-DoF quadrotor flight problem with combined \always{}, \implication{}, and \eventually{}-type specifications. The implementation is available at https://github.com/UW-ACL/TrajOpt_CT-STL.

math.OC↗

arXiv2026

Optimal Acceleration for Proximal Minimization of the Sum of Convex and Strongly Convex Functions

When minimizing the sum of a convex and a strongly convex function, or when finding the zero of the sum of a monotone operator and a strongly monotone operator, Chambolle and Pock (2010) and Davis and Yin (2015) proposed accelerated mechanisms that achieve an $\mathcal{O}(1/N^2)$ convergence rate for the squared distance to the solution, but the optimality of this rate was not established. In this work, we present Fast Douglas--Rachford Splitting (FDR), an accelerated method that improves the constants established in the prior works, and provide a complexity lower bound establishing that both the $\mathcal{O}(1/N^2)$ convergence rate and the leading-order constant of FDR's rate are optimal.

math.OC↗

arXiv2026

Transcription-Induced Failure Modes in 6-DOF Rocket Landing Trajectory Optimization

Solving optimal control problems via large-scale NLP solvers depends on discretizing continuous dynamics. Yet, this transcription step hides critical vulnerabilities-most notably truncation error and invariant drift-that can drive solvers toward dynamically infeasible or suboptimal trajectories. To expose these hidden failures, we introduce a problem- and transcription-agnostic adversarial objective that leverages the structure of local truncation-error bounds to aggressively amplify such defects. When applied to a 6-DOF rocket-landing problem, we reveal a stark reliability gap: of fourteen transcription methods tested, only three satisfy rigorous validation criteria. These results also expose a striking performance inversion: even in the absence of classical stiffness, a fourth-order implicit scheme (GL2) matches the fidelity of a sixth-order explicit method (RK6). Using B-series expansions and symplectic Runge-Kutta theorems, we isolate the specific truncation errors and quaternion-invariant drift responsible for these failures. Crucially, these theoretical vulnerabilities dictate operational performance: in practical lateral-divert scenarios, the implicit GL2 consistently outperforms the explicit RK6 in both end-to-end solve speed and robustness.

math.OC↗

arXiv2026

Smooth and Exact Parameterization of Continuous-time Signal Temporal Logic Specifications for Trajectory Optimization

This paper presents a smooth parameterization of continuous-time Signal Temporal Logic (CT-STL) specifications for nonconvex trajectory optimization that is sound and complete up to the accuracy of the underlying numerical integration scheme. CT-STL provides a natural framework for encoding rich temporal and logical task requirements, but existing trajectory-optimization formulations typically enforce such specifications only at discrete sampling nodes. In contrast, the proposed method evaluates specifications in dense time, thereby guaranteeing continuous-time satisfaction of always predicates, which is critical for path constraints such as obstacle avoidance, while eliminating the node-induced conservatism of eventually predicates by allowing satisfaction at any time within the prescribed interval. These two dense-time constructions also serve as the main building blocks for handling more general CT-STL formulas, including complex until specifications. Furthermore, the proposed parameterization resolves the locality and gradient-masking issues inherent in standard quantitative semantics, yielding a more favorable landscape for gradient-based solvers. Although dense-time evaluation introduces additional function evaluations during discretization, it also permits substantially coarser temporal grids without sacrificing safety or logical fidelity. This, in turn, reduces the dimension of the resulting nonconvex program, which is often the dominant factor in trajectory-generation cost. The numerical effectiveness and semantic exactness of the proposed framework are demonstrated on an agile quadrotor flight problem subject to a complex continuous-time until specification. The implementation is available at https:// github.com/UW-ACL/TrajOpt_CT-STL

math.OC↗

arXiv2026

QOCO-GPU: A Quadratic Objective Conic Optimizer with GPU Acceleration

We present a GPU-accelerated backend for QOCO, a C-based solver for quadratic objective second-order cone programs (SOCPs) based on a primal-dual interior point method. Our backend uses NVIDIA's cuDSS library to perform a direct sparse LDL factorization of the KKT system at each iteration. We also develop custom CUDA kernels for cone operations and show that parallelizing these operations is essential for achieving peak performance. Additionally, we refactor QOCO to introduce a modular backend abstraction that decouples solver logic from the underlying linear algebra implementations, allowing the existing CPU and new GPU backend to share a unified codebase. This GPU backend is accessible through a direct Python interface and through CVXPY, allowing for easy use. Numerical experiments on a range of large-scale quadratic programs and SOCPs with tens to hundreds of millions of nonzero elements in the KKT matrix, demonstrate speedups of up to 50-70 times over the CPU implementation.

math.OC↗

arXiv2025

A Proximal Method for Composite Optimization with Smooth and Convex Components

We introduce prox-convex for minimizing $F(x)=g(x)+h(C(x))+s(R(x))$, where $g$ and $h$ are convex, $C$ and $s$ are smooth, and each component of $R$ is convex (possibly nonsmooth). Here $g$ captures general convex objectives and indicator functions for convex constraints, while the composite template simultaneously models convex penalties on smooth features $(h \circ C)$ and smooth couplings of convex (possibly nonsmooth) features $(s \circ R)$. Each prox-convex step forms a convex subproblem by linearizing only the smooth maps while preserving the existing convex structure. The resulting subproblem is made strongly convex with the proximal metric $Q_k=μ_k I+H_k^+ \succ 0$ where $μ_k$ is adapted using an implicit trust-region strategy, and $H_k^+ \succeq 0$ is an optional curvature term for local acceleration. Under mild Lipschitz/smoothness and a per-coordinate monotone-or-smooth condition, we prove subdifferential regularity, derive two-sided quadratic model error bounds with explicit constants, and obtain sufficient decrease with $O(\varepsilon^{-2})$ complexity for driving the norm of the metric prox-gradient below $\varepsilon$. Furthermore, a local error-bound condition for $F$ guarantees a metric step-size error bound and hence local $Q$-linear convergence of the function values. Using the Taylor-like model framework of Drusvyatskiy, Ioffe, and Lewis, we show that every cluster point of the iterates is limiting-stationary; under our regularity conditions, this further implies Fréchet stationarity. The same framework also establishes robustness to inexact subproblem solves and justifies a model-decrease termination rule.

math.OC↗

arXiv2025

Continuous-time Constrained Funnel Synthesis for Incrementally Quadratic Nonlinear Systems

This paper presents a convex optimization-based framework for synthesizing time-varying controlled invariant funnels and associated feedback control around a given nominal trajectory for nonlinear systems subject to bounded disturbances. Nonlinearities are modeled using incremental quadratic constraints, including Lipschitz, L-smooth, and sector-bounded nonlinearities. Funnel invariance is ensured via a DLMI. Together with pointwise-in-time LMIs for state and input constraints, we formulate a continuous-time funnel synthesis problem. To solve it using numerical optimal control techniques, the DLMI is reformulated into a differential matrix equality (DME) and an LMI, where the DME acts as a funnel dynamics equation. We explore different formulations of these funnel dynamics. Continuous-time constraint satisfaction is addressed through two convex methods: one based on intermediate constraint-checking points, and another using a successive convexification method with subgradients to handle nondifferentiable maximum eigenvalue functions. Theoretical justification is provided for the existence of a measurable and integrable subgradient for the latter. The method is demonstrated on two numerical examples: the control of a unicycle and a 6-degree-of-freedom quadrotor for obstacle avoidance.

math.OC↗

arXiv2025

Funnel Synthesis via LMI Copositivity Conditions for Nonlinear Systems

Funnel synthesis refers to a procedure for synthesizing a time-varying controlled invariant set and an associated control law around a nominal trajectory. The computation of the funnel involves solving a continuous-time differential equation or inequality, ensuring the invariance of the funnel. Previous approaches often compromise the invariance property of the funnel; for example, they may enforce the equation or the inequality only at discrete temporal nodes and do not have a formal guarantee of invariance at all times. This paper proposes a computational funnel synthesis method that can satisfy the invariance of the funnel without such compromises. We derive a finite number of linear matrix inequalities (LMIs) that imply the satifaction of a continuous-time differential linear matrix inequality guaranteeing the invariance of the funnel at all times from the initial to the final time. To this end, we utilize LMI conditions ensuring matrix copositivity, which then imply continuous-time invariance. The primary contribution of the paper is to prove that the resulting funnel is indeed invariant over a finite time horizon. We validate the proposed method via a three-dimensional trajectory planning and control problem with obstacle avoidance constraints, and a six-degree-of-freedom powered descent guidance.

math.OC↗

arXiv2025

Onboard Dual Quaternion Guidance for Rocket Landing

The dual quaternion guidance (DQG) algorithm was selected as the candidate 6-DoF powered-descent guidance algorithm for NASA's Safe and Precise Landing -- Integrated Capabilities Evolution (SPLICE) project. DQG is capable of handling state-triggered constraints that are of utmost importance in terms of enabling technologies such as terrain relative navigation. In this work, we develop a custom solver for DQG to enable onboard implementation for future rocket landing missions. We describe the design and implementation of a real-time-capable optimization framework, called sequential conic optimization (SeCO), that blends together sequential convex programming and first-order conic optimization to solve difficult nonconvex trajectory optimization problems, such as DQG, in real-time. A key feature of SeCO is that it leverages a first-order primal-dual conic optimization solver, based on the proportional-integral projected gradient method (PIPG). We describe the implementation of this solver, develop customizable first-order methods, and leverage convergence-accelerating strategies such as warm-starting and extrapolation, to solve the nonconvex DQG optimal control problem in real-time. Finally, in preparation for an upcoming closed-loop flight test campaign, we test our custom solver onboard the NASA SPLICE Descent and Landing Computer in a hardware-in-the-loop setting. We observe that our algorithm is significantly faster than previously reported solve-times using the flight-tested interior point method-based subproblem solver, BSOCP. Furthermore, our custom solver meets (and exceeds) NASA's autonomous precision rocket-landing guidance update-rate requirements for the first time, thus demonstrating the viability of SeCO for real-time, mission-critical applications onboard computationally-constrained flight hardware.

math.OC↗

arXiv2025

Six-Degree-of-Freedom Aircraft Landing Trajectory Planning with Runway Alignment

This paper presents a numerical optimization algorithm for generating approach and landing trajectories for a six-degree-of-freedom (6-DoF) aircraft. We improve on the existing research on aircraft landing trajectory generation by formulating the trajectory optimization problem with additional real-world operational constraints, including 6-DoF aircraft dynamics, runway alignment, constant wind field, and obstacle avoidance, to obtain a continuous-time nonconvex optimal control problem. Particularly, the runway alignment constraint enforces the trajectory of the aircraft to be aligned with the runway only during the final approach phase. This is a novel feature that is essential for preventing an approach that is either too steep or too shallow. The proposed method models the runway alignment constraint through a multi-phase trajectory planning scheme, imposing alignment conditions exclusively during the final approach phase. We compare this formulation with the existing state-triggered constraint formulation for runway alignment. To solve the formulated problem, we design a novel sequential convex programming algorithm called xPTR that extends the penalized trust-region (PTR) algorithm by incorporating an extrapolation step to expedite convergence. We validate the proposed method through extensive numerical simulations, including a Monte Carlo study, to evaluate the robustness of the algorithm to varying initial conditions.

math.OC↗

arXiv2025

Discrete lossless convexification for pointing constraints

Discrete Lossless Convexification (DLCvx) formulates a convex relaxation for a specific class of discrete-time non-convex optimal control problems. It establishes sufficient conditions under which the solution of the relaxed problem satisfies the original non-convex constraints at specified time grid points. Furthermore, it provides an upper bound on the number of time grid points where these sufficient conditions may not hold, and thus the original constraints could be violated. This paper extends DLCvx to problems with control pointing constraints. Additionally, it introduces a novel DLCvx formulation for mixed-integer optimal control problems in which the control is either inactive or constrained within an annular sector. This formulation broadens the feasible space for problems with pointing constraints. A numerical example is provided to illustrate its application.

math.OC↗

arXiv2025

Successive Convexification for Passively-Safe Spacecraft Rendezvous on Near Rectilinear Halo Orbit

We present an optimization-based approach for fuel-efficient spacecraft rendezvous to the Gateway, a space station that will be deployed on a near rectilinear halo orbit (NRHO) around the Moon. The approach: i) ensures passive safety and satisfies path constraints at all times, ii) meets the specifications for critical decision points along the trajectory, iii) accounts for uncertainties that are common in real-world operation, such as due to orbital insertion, actuation, and navigation measurement, via chance constraints and utilizes a stabilizing feedback controller to bound the effect of uncertainties. We leverage sequential convex programming (SCP) and isoperimetric reformulation of path constraints, including passive safety, to eliminate the risk of inter-sample constraint violations that is common in existing methods. We demonstrate the proposed approach on a realistic simulation of a rendezvous to the Gateway.

math.OC↗

arXiv2025

Optimal Preconditioning for Online Quadratic Cone Programming

First-order conic optimization solvers are sensitive to problem conditioning and typically perform poorly in the face of ill-conditioned problem data. To mitigate this, we propose an approach to preconditioning--the hypersphere preconditioner--for a class of quadratic cone programs (QCPs), i.e., conic optimization problems with a quadratic objective function, wherein the objective function is strongly convex and possesses a certain structure. This approach lends itself to factorization-free, customizable, first-order conic optimization for online applications wherein the solver is called repeatedly to solve problems of the same size/structure, but with changing problem data. We demonstrate the efficacy of our approach on numerical convex and nonconvex trajectory optimization examples, using a first-order conic optimizer under the hood.

math.OC↗

arXiv2025

Impulsive Relative Motion Control with Continuous-Time Constraint Satisfaction for Cislunar Space Missions

Recent investments in cislunar applications open new frontiers for space missions within highly nonlinear dynamical regimes. In this paper, we propose a method based on Sequential Convex Programming (SCP) to loiter around a given target with impulsive actuation while satisfying path constraints continuously over the finite time-horizon, i.e., independently of the number of nodes in which domain is discretized. Location, timing, magnitude, and direction of a fixed number of impulses are optimized in a model predictive framework, exploiting the exact nonlinear dynamics of non-stationary orbital regimes. The proposed approach is first validated on a relative orbiting problem with respect to a selenocentric near rectilinear halo orbit. The approach is then compared to a formulation with path constraints imposed only at nodes and with mesh refined to ensure complete satisfaction of path constraints over the continuous-time horizon. CPU time per iteration of 400 ms for the refined-mesh approach reduce to 5.5 ms for the proposed approach.

eess.SY↗

arXiv2025

Revisiting Lossless Convexification: Theoretical Guarantees for Discrete-time Optimal Control Problems

Lossless Convexification (LCvx) is a modeling approach that transforms a class of nonconvex optimal control problems, where nonconvexity primarily arises from control constraints, into convex problems through convex relaxations. These convex problems can be solved using polynomial-time numerical methods after discretization, which converts the original infinite-dimensional problem into a finite-dimensional one. However, existing LCvx theory is limited to continuous-time optimal control problems, as the equivalence between the relaxed convex problem and the original nonconvex problem holds only in continuous time. This paper extends LCvx to discrete-time optimal control problems by classifying them into normal and long-horizon cases. For normal cases, after an arbitrarily small perturbation to the system dynamics (recursive equality constraints), applying the existing LCvx method to discrete-time problems results in optimal controls that meet the original nonconvex constraints at all but no more than $n_x - 1$ temporal grid points, where $n_x$ is the state dimension. For long-horizon cases, the existing LCvx method fails, but we resolve this issue by integrating it with a bisection search, leveraging the continuity of the value function from the relaxed convex problem to achieve similar results as in normal cases. This paper improves the theoretical foundation of LCvx, expanding its applicability to real-world discrete-time optimal control problems.

math.OC↗

arXiv2025

Newton-PIPG: A Fast Hybrid Algorithm for Quadratic Programs in Optimal Control

We propose Newton-PIPG, an efficient method for solving quadratic programming (QP) problems arising in optimal control, subject to additional set constraints. Newton-PIPG integrates the Proportional-Integral Projected Gradient (PIPG) method with the Newton method, thereby achieving both global convergence and local quadratic convergence. The PIPG method, an operator-splitting algorithm, seeks a fixed point of the PIPG operator. Under mild assumptions, we demonstrate that this operator is locally smooth, enabling the application of the Newton method to solve the corresponding nonlinear fixed-point equation. Furthermore, we prove that the linear system associated with the Newton method is locally nonsingular under strict complementarity conditions. To enhance efficiency, we design a specialized matrix factorization technique that leverages the typical sparsity of optimal control problems in such systems. Numerical experiments demonstrate that Newton-PIPG achieves high accuracy and reduces computation time, particularly when feasibility is easily guaranteed.

math.OC↗

arXiv2025

Ignition Point Reachability for Aerodynamically-Controlled Reusable Launch Vehicles

We describe a successive convex programming (Sequential Convex Programming (SCP)) based approach for estimate the set of points where a 5-degree of freedom (5-DoF) reusable launch vehicle (RLV) returning to a landing site can transition from aerodynamic to propulsive descent. Determining the set of feasible ignition points that a RLV can use and then safely land is important for mission planning and range safety. However, past trajectory optimization approaches for RLVs consider substantially simplified versions of the vehicle dynamics. Furthermore, prior reachability analysis methods either do not extend to the full constraint set needed for an RLV or are too beset by the curse of dimensionality to handle the full 5-DoF dynamics. To solve this problem, we describe an algorithm that approximates the projection of a high dimensional reachable set onto a low dimensional space. Instead of computing all parts of the reachable space, we only calculate reachability in the projected space of interest by using repeated trajectory optimization to sample the reachable polytope in the reduced space. The optimization can take into account initial and terminal constraints as well as state and control constraints. We show that our algorithm is able to compute the projection of a reachable set into a low dimensional space by calculating the feasible ignition points for a two-phase aerodynamic/propulsive RLV landing trajectory, while also demonstrating the aerodynamic divert enabled by our body and fin actuator model.

eess.SY↗