Influence and Optimization Horizon

• Influence and Optimization Horizon is a framework that defines how planning windows and intervention timing impact strategy robustness across various fields.
• The topic underscores methodologies in portfolio selection, control theory, and network influence, emphasizing continuous convergence and adaptive control strategies.
• Research shows that optimizing horizon length improves computational efficiency and long-term outcomes in dynamic and uncertain environments.

Influence and Optimization Horizon

The concepts of influence and optimization horizon arise across disciplines such as mathematical finance, control theory, combinatorial optimization, and the modeling of influence in social and economic networks. These notions formalize how the choice of horizon—be it a time period, planning window, or structural extent—determines the effectiveness and robustness of optimal strategies, especially under uncertainty or evolving system dynamics. The optimization horizon sets the scope for which an objective is optimized, while influence structures, both endogenous and exogenous, dictate how interventions propagate and impact outcomes within this horizon.

1. Horizon Dependence in Utility Maximization

In incomplete financial markets modeled by multi-dimensional Brownian motion, the optimization horizon refers to the time $T$ at which the expected utility of terminal wealth is maximized via portfolio selection. The underlying wealth process $X_t$ evolves according to

$$dX_t = X_t r_t dt + X_t \pi_t \bigl[(\mu_t - r_t) dt + \sigma_t dB_t\bigr],$$

where $r_t$, $\mu_t$, and $\sigma_t$ represent interest rates, asset drifts, and volatilities, and $\pi_t$ is the portfolio process.

A fundamental result is that, given appropriate regularity and finiteness conditions (e.g., $u^{(T+\epsilon)}(x) < \infty$ for some $\epsilon > 0$ and all $x > 0$), both the value function $u^{(T)}(x)$ and the optimal terminal wealth depend continuously on the investment horizon $T$. Concretely,

$$\lim_{K \uparrow T} u^{(K)}(x) = u^{(T)}(x), \qquad \forall x > 0,$$

as established in Theorem 3.1 and strengthened by Theorem 3.5, which also provides uniform convergence on compacts.

However, this continuity may fail when a $T$-horizon-optimized strategy is terminated prematurely at time $K < T$. Defining $u^{(T)}(K,x) = \mathbb{E}[U(X^{(T)}_K)]$, the paper constructs examples where $\lim_{K \uparrow T} u^{(T)}(K, x)$ does not converge to $u^{(T)}(x)$—particularly for negative power utilities. This non-convergence is rooted in strategies that defer gains to the terminal moment, leading to low or nearly zero interim wealth, and catastrophic utility loss if forced to exit early.

Addressing this, necessary and sufficient conditions are given to preclude non-convergence. For instance, if the utility function meets boundedness and marginal utility control conditions, or if the market price of risk satisfies a uniform exponential integrability condition, then

$\lim_{K \uparrow T} u^{(T)}(K, x) = u^{(T)}(x)$

is ensured. These safeguard conditions are essential for robust strategy design in the face of potential horizon uncertainty.

2. Optimization Horizon in Control and Planning

In model predictive control (MPC), the optimization horizon is the number of future steps $N$ the controller considers when optimizing control actions. The horizon length directly affects closed-loop suboptimality, computational load, and stability guarantees.

Strategies have been developed to adaptively adjust $N$ at each time step, balancing computational resources and lower bounds on suboptimality. Online algorithms compute the open-loop control sequence, evaluate a local suboptimality measure $\alpha(N)$, and adaptively prolong or shorten the horizon to guarantee performance. The relaxed Lyapunov inequality,

$$V_N(x(n)) \ge V_N(x(n+1)) + \alpha \ell(x(n), \mu_N(x(n))),$$

ensures that the finite-horizon policy closely tracks true infinite-horizon optimality, with $\alpha$ quantifying the suboptimality. Advanced methods leverage fixed-point mappings (e.g., $\Phi$ mapping in equation (1)) to compute suitable horizons in real time.

For trajectory planning in highly dynamic environments, receding horizon (or rolling horizon) optimization solves local, shorter-horizon problems iteratively. The window length $T$ is tuned to balance trajectory quality and computation time, leveraging terminal constraints (often derived from a precomputed nominal trajectory) to ensure recursive feasibility and convergence to the terminal state. Empirical results show significant improvements in cost and execution time when $T$ is appropriately chosen.

3. Influence Propagation and the Optimization Horizon in Networks

Optimizing influence in social or economic networks involves selecting nodes or channels to maximize the reach or impact within a specified time or structural horizon. Analytical models employ both exogenous (directly controlled) and endogenous (network-diffused) influence mechanisms.

Optimal control analysis reveals “bang-bang” solutions: at any moment, channels are used maximally or not at all, with bounded switching times. The influence of each channel or node evolves over the campaign or intervention horizon—the allocation favors broad reach (wide audience) at the outset, shifting to targeted interventions (late-deciders, opinion leaders) near the terminal event (e.g., election day) (1702.03432).

Novel frameworks, such as cycle-based influencer selection, depart from traditional node centrality ranking by analyzing the roles of cycles—mesoscale structures essential for efficient influence spread. By ranking basic cycles with indicators covering community participation, transmission paths, and local clusters, the cycle ranking method (CycRak) selects more dispersed and structurally effective influencers, expanding the dissemination range up to threefold compared to hub-focused methods (2405.09357).

A key insight in random graph models (e.g., independent cascade/SIR processes) is the existence of a narrow “optimization horizon” in parameter space: only near percolation critical points does optimization markedly outperform random seed selection. As network size grows, this horizon shrinks as a power law, questioning the utility of global optimization except within vanishingly narrow parameter regimes (1708.02142).

4. Horizon Effects in Learning and Meta-Optimization

The optimization horizon is central in meta-optimization, notably in tuning hyperparameters (learning rates, schedules) for neural network training. Due to the expense of backpropagating through long training runs, meta-objectives are often defined over much shorter horizons.

This truncation induces “short-horizon bias”: optimizers learn to favor conservative, short-term improvements (smaller step sizes), even if such choices are suboptimal in the long run (1803.02021). Analytical and empirical results demonstrate that the bias persists for horizon lengths typical of meta-optimization (e.g., 100 steps) and that longer horizons—while costly—are essential to recover long-run optimality. Mitigating the bias may require novel meta-objectives or approximations that account for longer-term effects.

5. Investment Horizon and Uncertain Timing in Portfolio Optimization

The horizon dependence of utility optimization deepens in the presence of horizon uncertainty and non-concave utilities. When the investment horizon $\tau$ is a random variable, and the utility function is not concave, standard methods (e.g., concavification) may fail to yield true optimality unless $\tau$ is a stopping time with respect to the market filtration. If $\tau$ is independent of market risk, the expected utility can be strictly suboptimal, and a recursive procedure based on dynamic programming is proposed to recover optimality (2005.13831).

This scenario generates multimodal wealth distributions—multiple local maxima emerge due to the combination of non-concavity and horizon randomness—conveying the investor a certain flexibility to switch between local maxima as market conditions realize. The optimal solution requires a martingale condition on the candidate wealth process, with a weighted sum of the Lagrange multipliers constant across possible stopping dates—captured by a system of equations relating the inverse marginal utility.

6. Specialized Horizons in Time-Dependent and Safety-Critical Control

Some control frameworks distinguish between the prediction horizon (over which performance is optimized) and the constraint horizon (over which state or safety constraints are enforced) (2503.18521). By relaxing constraints in the later stages of the optimization while maintaining a performance-oriented prediction horizon, the controller can achieve a trade-off: guarantee immediate safety and recursive feasibility but allow more flexible, less myopic, and less conservative actions that enhance long-term system performance.

A mathematical quantification of closed-loop suboptimality is given via parameters such as

$$\alpha = 1 - \frac{\beta^{N - \bar{N} + 1}}{(\beta + 1)^{N - \bar{N} - 1}},$$

where $N$ is the prediction horizon and $N - \bar{N}$ the number of constrained steps. As the ratio increases (more constraint enforcement), $\alpha$ approaches 1 and closed-loop performance aligns closely with open-loop optimality.

This dual-horizon design has implications for the integration of stability (via Control Lyapunov Functions, CLFs) and safety (via Control Barrier Functions, CBFs): by planning over longer prediction horizons while enforcing CBFs only where necessary, the scheme reduces incompatibility and myopic behaviors endemic to one-step QP-based CBF methods.

7. Broader Implications and Horizon-Aware Optimization Across Domains

Across stochastic programming and simulation-optimization in production planning, the rolling horizon paradigm—periodically solving optimization problems with updated information—incorporates the latest uncertainty forecasts and system states (2402.14506). Scenario-based formulations within rolling horizons allow adaptive, cost-efficient, and robust decisions, with horizon length and flexibility (number of periods before decisions are “frozen”) trading off against aggregate costs and service levels in uncertain production environments.

In combinatorial optimization, learning-guided rolling horizon frameworks combine heuristic optimization with machine learning predictors to aggressively fix variables deemed stable across subproblem boundaries, reducing redundant computation and improving solution quality, particularly in long-horizon flexible job-shop scheduling (2502.15791).

In all these contexts, the selection and adaptation of the optimization horizon—sometimes fixed, sometimes dynamic and learning-driven—affects not just computational tractability and policy effectiveness, but the very structure of the optimal solution, the robustness to misspecification and uncertainty, and the capacity to anticipate and react to the dynamic nature of modern decision-making environments.