Papers

We propose a sequential smoothing majorant stochastic approximation (SMSA) method for nonconvex-nonconcave minimax optimization problems. To overcome the nonconcavity of the inner maximization problem, we introduce a new majorant stochastic approximation with an $O\left(\beta_N^2\right)$ accuracy bound, where $\beta_N$ is the sample coverage radius. The accuracy bound significantly improves upon the $O\left(\beta_N\right)$ approximation bound for the standard stochastic approximation. Moreover, we establish nonasymptotic bounds for both global optimal values and minimizer sets, and prove consistency for the Clarke stationary points, as $\beta_N \downarrow 0$ almost surely. We show that the generated sequence by the SMSA method is bounded, and that the returned point is an approximate Clarke stationary point of the majorant stochastic approximation models. Numerical experiments on a synthetic toy example and robust logistic regression on two UCI datasets demonstrate improved approximation fidelity and lower mean robust test losses relative to the standard sampled approximation.

Characterizing all PID gains that guarantee uniform exponential regulation of uncertain nonlinear systems remains a fundamental problem. Most existing results provide only sufficient conditions derived from a particular Lyapunov construction. This paper gives exact gain regions, defined directly by uniform regulation over the entire uncertainty class, for two classes of nonaffine MIMO systems with $m\ge2$ controlled coordinates and input channels. For systems of second order under PID control, we independently characterize a common quadratic inner region and an outer region based on a linear subclass. The two regions coincide and therefore characterize the PID gain region exactly at the system level. For systems of first order under PI control, a lossless reduction to a linear family on the uncertainty boundary, combined with a common quadratic certificate, yields the exact PI region. Both regions strictly enlarge existing sufficient regions, and the PID result also provides a sequential tuning rule. A compact linear family with one uncertain parameter shows that common quadratic certification is not generally lossless. These results establish endpoint coincidence and boundary family equivalence as two routes to exact gain regions and motivate the broader question of which nonlinear uncertainty classes admit such characterizations.

In this work, we study the null controllability by one-sided boundary controls of one-dimensional hyperbolic balance laws with constant coefficients. Our first result shows that, when the system has only one negative or positive speed, the minimal null control time of such systems depends on some orthogonality conditions for a particular sequence. This sequence is explicit in function of the coefficients of the system but it is defined by a nonlinear recurrence relation. Our second result then completes the previous one by giving explicit bounds on the number of orthogonality conditions that have to be checked in two nontrivial situations. The proofs rely on a careful analysis of the so-called kernel equations associated with the system, including a new well-posedness result. Our results are also valid for the finite-time stabilization property.

We introduce cooperative integer programming games (CIPGs), in which agents pool budget constraints to accomplish indivisible tasks jointly and the characteristic function maps every coalition to the optimal value of a pooled integer program. Our goal is to identify an optimal coalition structure (OCS) and a stable one (OSCS). We derive a stability inequality that keeps each formed coalition in the Core with respect to itself, and present two mixed-integer OCS formulations, aggregated and disaggregated, proving that the disaggregated formulation is integer-equivalent yet yields a tighter LP relaxation. Building on the stability inequality we develop lifted stability cuts, several separation strategies inside a cutting-plane algorithm, an SCS-feasible primal heuristic that constructs warm starts with guaranteed stability, and a payoff-refinement step computing the Shapley value and the nucleolus of every formed coalition. On benchmark cooperative knapsack games, the method certifies optimality with up to 16 players and reaches MIP gaps below 1% at 30 players while evaluating 766 of the roughly $10^9$ coalition values.

We extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is, for the first time, the ability to consider more than two sets, a task which has been in the mind of researchers for many years without much success in fulfilling it because of a certain theoretical obstacle related to cycles and minimizers of general functionals. The other direction is the ability to handle each set as an intersection of a finite family of sets. We call the resulting problem the "Best Approximation Tuple (BAT) problem". The fundamental observation that leads to these extensions is to recognize and handle one set (the "pivot set") as different from the remaining sets (the "satellite sets") instead of seeking cycles as the minimizers of a target functional. We prove the convergence of the algorithm to the unique solution of the problem in the Euclidean case with strictly convex and compact satellite sets. Because of the lack of Fejér monotonicity, our convergence analysis is not standard, and is based on almost unknown properties of orthogonal projections regarding equality and inequality in the definition of nonexpansiveness.

Data-driven distributionally robust optimization (DRO) typically treats the conditional outcome law as fixed and uses ambiguity sets to capture estimation error. This paper studies latent distributional heterogeneity, where each instance has an unobserved law but contributes only one observation, so uncertainty persists even if the mixture law is known. We propose Conformal-DRO, which uses nested conformal regions to construct an ambiguity set for the future latent law. Under exchangeability, the set covers this law with probability at least $1-\alpha$ in finite samples, without estimating underlying latent laws or their mixing mechanism. The conformal path induces a data-driven transport geometry, while $\alpha$ determines the radius. The worst-case problem reduces to a finite linear program over conformal shells and admits sparse adversarial solutions. The resulting robust value provides a finite-sample certificate for the selected decision's expected cost.

Dynamic allocation decisions couple present resource use to evolving internal conditions, delayed returns, and future costs. We represent this interaction by four probability localizations linked through regime-indexed, graph-constrained column-stochastic operators. Pre-action state or context selects a locally affine model, while action-dependent changes update subsequent regimes, yielding a causal switched representation of nonlinear evolution. We characterize operator identifiability relative to the graph, the stochastic constraints, and the sampled embedding, separating coefficient recovery from predictive equivalence on the decision domain. Decision making is then formulated through implementable return–cost acceptability regions. Finite-horizon error propagation supplies conservative classification margins, and simultaneous intervals distinguish model-relative near-optimality from certified $\epsilon$-optimality over a declared finite policy class. Regime-indexed stochastic feedback is admitted when it satisfies the same certification test. Reproducible synthetic laboratories for personal preparation, supplier participation, and customer retention illustrate exact, operator-supplied, and noisy feedback cases. Multinomial experiments show improving recovery of the feedback function and fewer unresolved decisions with increasing sample size, while unrestricted off-policy recovery remains limited. The contribution is a structure-preserving representation–identification–decision workflow, not a domain-specific physiological or commercial calibration.

EGGROLL (Sarkar et al., 2026) makes evolution strategies (ES) practical for LLMs by replacing dense Gaussian weight perturbations with low-rank Gaussian products, often of rank one. This choice is computationally attractive but geometrically severe: Each rank-one perturbation lies in a zero-volume subset of the ambient matrix space, despite having identity covariance. We characterize the EGGROLL update mean field at finite rank and nonzero perturbation radius as a resolvent applied to the gradient of the perturbation-smoothed objective. This transformation can make the mean field nonconservative and reverse the local stability of an optimum. EGGROLL nevertheless recovers the gradient exactly on quadratic objectives at every rank and radius. In finite populations, the additional sampling variance of rank-one perturbations relative to dense Gaussian ES decays inversely with matrix width under a local affine model, and is only $0.098\%$ at width $4096$. Finally, we introduce LOO-ROLL, a leave-one-out estimator that replaces EGGROLL's two antithetic evaluations per direction by one. At equal evaluation cost, LOO-ROLL halves estimator MSE in transformer blocks. Across fourteen post-training settings up to 14B parameters, matched-time comparisons with EGGROLL yield eleven improvements in individual paired tests and no significant loss. At 1.7B, 8B, and 14B parameters, matched-time gains are $2.9$, $14.1$, and $7.9$ percentage points on GSM8K and $12.2$, $8.4$, and $8.1$ points on MATH-500.

Minimum-distance bipartite matching on general networks has numerous applications various fields. This paper first focuses on deterministic problems and presents an exact edgewise-separable convex-flow reformulation. By introducing a smooth monotone rearrangement approximation of the edge-wise imbalance profiles, the convex-flow reformulation's can be solved efficiently. If we further conduct a first-order resistance-based approximation of the convex program, a one-step Laplacian-based estimator can be analytically derived in closed forms. The paper also studies random problems where supply and demand points are randomly distributed. We show that the expected optimal matching distance scales with the square root of the number of points if the supply/demand point distributions are identical, or linearly otherwise. In the former case, the optimal flow is proven to be centered, symmetric, and sub-Gaussian. In the latter case, the limiting resistance network characterizes how supply-demand imbalance is redistributed and motivates a fast algorithm that approximate the optimal flow based on the limiting resistance. Numerical experiments show that the proposed estimators closely approximate the exact matching cost while substantially reducing computation time. The proven theoretical properties of the random matching solution are numerically verified by large-scale Monte Carlo simulations.

In this paper, we establish some new localization sets for Pareto Z-eigenvalues of tensor eigenvalue complementarity problems and discuss the relationships among existing ones. As applications, a sufficient condition for the solution existence of the tensor complementarity problem (TCP) and the tensor generalized eigenvalue complementarity problem (TGEiCP) are presented. In addition, this condition is also used to test the vacuum stability of a general scalar potential

We study the maximum-volume inscribed ellipsoid (MaxIE) problem for a polytope through a geometric iteration based on polarity. Given an interior point, the method forms the shifted polar polytope, computes its minimum-volume covering ellipsoid (MinCE), and then polarizes this covering ellipsoid back to obtain a new inscribed ellipsoid. This procedure was suggested by Khachiyan and Todd in 1993. Prior work studied basic polarity identities and gave an asymptotic-convergence argument for the exact process, but did not furnish a volume-ratio rate. Independently of that asymptotic argument, we develop a new analysis based on the log-volume of the polar MinCE as a potential function. We prove its convexity with an explicit gradient formula and establish global linear contraction of the potential gap along each trajectory. The contraction factor is existential and instance-dependent. This gives both a separate convergence proof and a finite volume-ratio iteration bound. We further analyze an inexact polarity process in which each MinCE subproblem is solved only approximately, and derive sufficient oracle tolerances for producing a prescribed volume approximation. Combining this outer analysis with existing algorithms for MinCE gives conditional arithmetic estimates based on the path-following Newton method, the barycentric coordinate descent, and the away-step Frank-Wolfe method. Numerical experiments compare these oracle choices with two MaxIE baselines and illustrate that performance depends on matching the MinCE solver to the instance geometry.

In this note, we provide a short and direct proof that approximately solving $Ax=b$ to relative error $\varepsilon$, where $A$ has condition number $\kappa$ and unrestricted dimension, requires $\Omega(\kappa\log(1/\varepsilon))$ matrix-vector multiplications in the worst case, even for randomized algorithms. This essentially recovers the lower bound of Dereziński, Epperly and Meyer [2026] for this setting, whose elegant and more general approach inspired us to seek a short direct proof. A straightforward reduction implies the classical $\Omega(\sqrt{\kappa}\log(1/\varepsilon))$ lower bound for optimizing strongly convex quadratic functions, applicable to randomized algorithms.

Reinforcement learning (RL) agents deployed in real-world environments are often vulnerable to adversarial perturbations in state observations, creating risks in safety-critical applications. Certification methods can improve robustness against adversarial perturbations by providing lower bounds on expected cumulative rewards. Existing certification methods, however, mainly focus on risk-neutral objectives. In this paper, we extend certification methods to risk-sensitive objectives by establishing lower bounds on the exponential utility of cumulative rewards under $l_{p}$-norm-bounded state adversarial perturbations ($1\leq p <\infty$). By introducing a $\phi$-divergence relaxation of the perturbation set, we formulate the risk-sensitive certification problem as a convex optimization and derive its dual to obtain a tractable approximation of the certified lower bound. We further propose an empirical method that improves certified lower bounds by selecting the training risk-aversion parameter $\beta$ independently of the risk level used during evaluation. Experiments on both OpenAI Gym environments and a machine replacement problem show that, compared to risk-neutral training, risk-averse training generally yields policies with higher certified lower bounds, particularly under larger perturbation budgets. Moreover, under both risk-neutral and risk-averse evaluation settings, increasing risk aversion during training leads to non-monotonic certification performance, where certified lower bounds initially improve but eventually decrease due to overly conservative policies.

A method for constrained computation guidance and control is described. The method employs guidance updates by periodically re-solving a constrained optimal control problem. Feasibility of the optimal control problem is improved by preventing constraint violations in the perturbed system. Constraint violation is prevented by augmenting the optimal control problem with a penalty term intended to increase the margin between a constraint and its limit when the perturbed system is within certain proximity of the constraint limit. The additional penalty term employs a smooth maximum approximation to increase the margin between the constraint and its limit. The method developed in this paper is demonstrated on the maximum cross-range reusable launch vehicle (RLV) entry problem, for which the reference solution contains a long-duration constrained arc. The results demonstrate that the method developed in this paper is viable for path-constrained optimal guidance and control.

Water distribution networks (WDNs) are susceptible to various failures, including but not limited to human errors, cyber-attacks, and network modifications, necessitating the vulnerability analysis of WDNs. Graph-theoretic centrality measures—as a main class of centrality measures—aim to rank the network components solely based on their influence (i.e., criticality) on the WDN topology in the case of input changes, while overlooking the WDN dynamics. To overcome such a limitation, this paper uses a control-theoretic centrality measure to identify the network's most and least influential pipes in WDNs, by simultaneously incorporating the dynamics and topology of the WDN. First, given a WDN modeled by nonlinear differential-algebraic equations (NDAEs) consisting of transient flow dynamics as differential equation (DE) and conservation of water mass as algebraic equation (AE) and considering the pipe flow rates as states of the state-space (SS) representation, we extract a linearized system modeled by linear ordinary differential equations (LODEs) around the equilibrium (i.e., steady) flow rate vector. Second, treating pipe flow rates as SS nodes (states), we introduce a node centrality-based measure, namely vulnerability vector (VV), to rank the network pipes based on their influence on the dynamics and topology of the WDN in the case of input changes. In particular, the network's most and least influential pipes can be identified through such a centrality-based approach. This enables water engineers to understand the WDN's vulnerability better and effectively prioritize the maintenance and operational efforts on the most influential pipes within the WDN.

This paper develops a principled framework for automatically tuning preconditioners in operator splitting methods, including forward-backward splitting, Douglas-Rachford splitting, and the alternating direction method of multipliers. We interpret these splitting methods as preconditioned gradient descent applied to their corresponding envelope functions and use online learning to adaptively learn the preconditioner during optimization. Our framework achieves asymptotically faster convergence rates and accelerates the solution of important optimization problems, including Lasso, $\ell_1$-regularized logistic regression, and quadratic programming. Extensive numerical experiments validate our theoretical results and demonstrate the practical effectiveness of the proposed approach.

Motivated by Wasserstein measures of dependence, we study the largest possible 2-Wasserstein distance between a joint distribution and the product of its prescribed marginals. For two uniform marginals, Catalano and Lavenant conjectured that the monotone and antimonotone couplings maximize the distance from the independent coupling. We prove the Gaussian analogue for an arbitrary number $n\geq 2$ of one-dimensional standard Gaussian marginals. More generally, for every probability measure $\mu$ on $\mathbb R$ with finite second moment, we characterize the laws on $\mathbb{R}^n$ with all marginals equal to $\mu$ that are farthest from the $n$-dimensional standard Gaussian.

We analyze a stochastic Newton optimization scheme for locating the unique global minimizer of a general nonconvex objective function. The method couples a Newton algorithm to additive Gaussian noise with state-dependent variance. In the bounded domain setting, we prove global almost sure convergence. The proof is based on two features of the algorithm: a nondegenerate exploratory property that ensures entrance into a neighborhood of the minimizer after a finite number of steps, and a decaying-noise property that yields contraction with high probability and prevents infinitely many exits from the neighborhood of the minimum.

We study convex quadratic minimization over the unit $L_1$ ball in which the maximum eigenvalue of the Hessian matrix is bounded by a positive constant $L$. We propose a novel first-order algorithm with objective value error bounded by $O(L/T^3)$ after $T$ gradient evaluations, assuming that the subproblems involved in the algorithm can be solved exactly. To the best of our knowledge, the best convergence rate of algorithms in the literature is $O(L/T^2)$. From the perspective of information-based complexity theory, our proposed algorithm is the first in the literature that achieves the $O((L/\varepsilon)^{1/3})$ first-order oracle complexity, although its current version is not necessarily practical for implementation. We hope that our proposed algorithm could shed some light on future implementable and efficient $O(L/T^3)$-convergence-rate algorithms. The proposed algorithm incorporates a decomposition of components of vectors in the unit $L_1$-norm ball to "good" and "bad" parts, and uses symmetric rank-1 (SR1) updates on the bad parts. The proposed algorithm was developed after the author instructed the OpenAI ChatGPT 6 (Astra) model to study the problem using ideas of weak-type $L^1$ estimates and good-bad part decomposition in harmonic analysis and a recent result.

Exact algorithms for solving mixed-integer bilevel linear programs (MIBLPs) typically rely on sequences of lower and upper bounds that converge to the optimal value. These procedures are commonly initialized using the single-level relaxation (SLR), obtained by omitting the follower's optimality condition and solving the resulting single-level optimization problem. In this paper, we investigate whether, for broad classes of MIBLPs, the resulting standard bounds admit uniform improvements that can be computed within the same computational complexity regime. For pure continuous bilevel linear programs, we show that, unless $P = NP$, neither the SLR-based lower bound nor its associated upper bound can be uniformly improved in polynomial time, even for the class of min-max problems. We then extend this analysis to the class of pure integer min-max bilevel linear programs under the assumption that the polynomial hierarchy does not collapse. First, we show that the continuous relaxation of the SLR admits no uniform polynomial-time computable improvement. We then prove that neither the SLR itself nor its associated upper bound admits a uniform improvement by a polynomial-time algorithm with access to a mixed-integer linear programming (MILP) oracle. Importantly, this rules out uniform improvements by iterative MILP-based approaches, including cutting-plane-based and decomposition algorithms. Overall, our results demonstrate that the SLR-based bounds are, in a complexity-theoretic sense, unimprovable systematically within their natural computational regimes.

Low-voltage (LV) distribution feeders increasingly require internal state information, yet real-time telemetry is often limited to feeder-head active and reactive power, while downstream demand and production are described only by static contract metadata. This paper proposes a physics-based AC circuit-embedding reconstruction framework for this extreme-sparsity regime. Feeder-head measurements, matched metadata, and class-level profiles first determine bus-level load and production priors. These priors parameterize auxiliary constant-power elements coupled to the physical feeder through impedance branches, and the feeder state is reconstructed by solving the resulting nonlinear AC circuit equilibrium. The resulting AC equilibrium enforces the assigned virtual-element powers, while weighted least squares (WLS) reconciles measurements and pseudo-measurements through weighted residual fitting. The method is evaluated on a real 459-bus Portuguese LV feeder model using an in-sample-calibrated synthetic benchmark of 672 fifteen-minute snapshots. The proposed method converged for all 672 snapshots and achieved voltage RMSE, MAE, and maximum absolute error of 0.0033, 0.0014, and 0.0337 p.u., respectively, outperforming both static-prior WLS baselines. Under the same constructed priors, matched-prior WLS achieved a lower RMSE of 0.0029 p.u., while the proposed circuit embedding reconstructs the feeder state by solving the nonlinear AC equilibrium, bypassing WLS residual fitting entirely and requiring no specification or tuning of pseudo-measurement variances.

We introduce Selector-Anchored Diffusion with Local Innovation (D+LI), a distributed vector-assembly primitive that constructs complete local replicas from independently owned authoritative blocks. Orthogonal selectors preserve block identity, local innovation anchors each owner's contribution, and Laplacian diffusion propagates these contributions across the network. We show that pure averaging with raw additive selector sources cannot isolate this replica. The D+LI error operator decomposes coordinatewise as the identity minus an owner-grounded Laplacian, yielding global contraction with a factor independent of the total and block dimensions. For time-varying outer targets, we establish a finite-T input-to-state stability (ISS) bound and a self-consistent estimator-controller gain condition for Lipschitz coordination maps, including non-potential ones. For semi-algebraic potential problems, we prove Kurdyka-Lojasiewicz (KL) finite-length convergence of the coupled decision-assembly iteration with a fixed finite communication budget per outer iteration under explicit timescale-separation conditions. Three numerical experiments demonstrate contraction tightness and finite-horizon tracking, verify an admissible KL timescale, and illustrate discrete and non-potential outer maps.

This document describes two developments. The first is a Multi-stage Decision Framework that permits the formulation of various sequential decision-making problems in a generic manner. The second concerns a robust stochastic solution procedure designed to handle the binary nonlinear optimization problem stemming from the proposed decision framework. The method adopts Monte-Carlo search with incremental probabilistic learning with reinforcement. The two developments are described herein along with a demonstrative application example. Key words: Multi-stage, decisions, binary, nonlinear systems, probabilistic and reinforcement learning

We study the one-corotational harmonic map heat flow from the disk to the sphere with a Dirichlet boundary control. We introduce a control-adapted gluing strategy to prevent blow-up. More precisely, we construct explicit upper profiles and well-prepared boundary traces such that every nonnegative datum below one of these profiles and having boundary value above $\pi$ blows up under the corresponding constant trace, whereas the corresponding solution is global under every control below the associated well-prepared boundary trace. Controls with uniformly bounded time derivative also prevent infinite-time concentration. We also show that boundary control can drive any nonnegative datum to blow-up before any prescribed time, whereas bounded controls cannot prevent suitably fast concentration.

We consider the observability and controllability for the Schrödinger equation on polyhedra. We show that for the Schrödinger equation on convex polyhedra in $\mathbb{R}^d\,(d\geq2)$, when the initial data possess $H^s\,(s>d/2)$ regularity, observability and controllability hold, with the control region being an arbitrary nonempty open neighborhood of the singular set of the boundary, which is the same as the control region for eigenfunctions. Our proof of observability is a concrete adaptation of the method in the Burq–Zworski black box control work, where the observability inequality is proved via a resolvent estimate. Hence, before proving observability, we present an observability resolvent estimate on polyhedra. The idea of the proof of this resolvent estimate comes from the case of eigenfunction concentration on polyhedra considered by Cekić–Georgiev–Mukherjee, which uses semiclassical measures and relies closely on the dynamical properties of the billiard flow in polyhedra. Finally, by a standard HUM argument, we show that observability implies controllability.

The 11/6 worst-case constant of the Wang-Sitters rounding scheme, which a companion note establishes, can naturally be attributed to the freedom in Step 3, where an arbitrary valid slot matching is permitted. We show that eliminating that freedom does not improve the constant. A minimum-makespan completion oracle still has worst-case constant exactly 11/6 against the optimum; both natural 7/4 statements about it are false; and restricting Step 1 to vertices of the relaxation does not help. The loss therefore cannot be attributed solely to the freedom in Step 3. We also record what structure survives: a reduction confining every overload to two shapes, a seven-machine instance defeating the natural two-phase repair, and a strict 7/4 bound on the generalized three-path family.

This paper focuses on the boundary approximate controllability of two classes of linear parabolic systems, namely a system of $n$ heat equations coupled through constant terms and a $2 \times 2$ cascade system coupled by means of a first order partial differential operator with space-dependent coefficients. For each system we prove a sufficient condition in any space dimension and we show that this condition turns out to be also necessary in one dimension with only one control. For the system of coupled heat equations we also study the problem on rectangle, and we give characterizations depending on the position of the control domain. Finally, we exhibit a cascade system for which the distributed controllability holds whereas the boundary controllability does not. The method relies on a general characterization due to H.O. Fattorini.

In this work we study the null-controllability properties of linear parabolic systems with constant coefficients in the case where several controls are acting on different distributed subdomains and/or on the boundary. We prove a Kalman rank condition in the one-dimensional case. In the case where only distributed controls are considered we also establish related results such as a Carleman estimate.

Modern large language models (LLMs) can translate natural-language descriptions into operations research (OR) formulations. Post-training techniques including reinforcement learning and on-policy self-distillation have further improved this capability. However, three limitations remain in training LLMs for OR formulations. First, training commonly relies on synthetic formulations validated by human experts or stronger models, constraining scalable supervision. Second, credit assignment is either coarse or costly: outcome rewards score an entire trajectory without locating the responsible modeling decision, whereas process-level supervision requires an additional evaluator. Third, privileged self-distillation can induce style mismatch by using solver context unavailable at deployment. We find that a model can improve from solver-artifact feedback generated by its own rollouts, making self-distillation a practical, evaluator-free source of dense supervision. Therefore, we propose SOLID: Solver-Informed On-Policy LearnIng through Self-Distillation, a novel framework for self-improving OR language models without verified answers or external evaluators. SOLID executes candidate programs from multiple rollouts, clusters their objectives, and selects a majority-group artifact as a pseudo-reference. The model then performs updates using group-relative advantages and dense self-supervision signals. Across multiple OR benchmarks, SOLID improves solution accuracy for both general-purpose and OR-tuned models over outcome-only group-relative training. These results show that solver artifacts can support scalable self-improvement without trusted answers.

We study optimal boundary mixing of a nondiffusive scalar transported by a two-dimensional incompressible second-grade fluid. The control is the tangential traction in a Navier-slip boundary condition, and the principal objective is the terminal $(H^1(\Omega))'$ mix-norm, supplemented by a quadratic control cost and an optional enstrophy reward. The second-grade constitutive law introduces a spatially filtered acceleration and a generalized vorticity; after lifting the nonhomogeneous boundary data, both $\mathbf{g}$ and $\partial_t\mathbf{g}$ enter the state equation, where $\mathbf{g}$ is the boundary control variable. We prove global state well-posedness for a passive scalar and well-posedness on a common, control-independent local interval for an active scalar. We then establish existence of an optimal control, directional differentiability of the control-to-state map in the topology required by the terminal objective, and a weak backward adjoint formulation. A duality identity yields the first-order variational inequality. In the absence of the friction coefficient $\beta$ and enstrophy reward weight $\zeta$ (that is, $\beta=\zeta=0$), the additional adjoint regularity implies uniqueness of the optimal control for all sufficiently large control penalties $\gamma$.