• Extend optimal experimental design for guaranteed parameter estimation to LLE systems • Employ rigorous calculations for the LLE using the Baker’s criterion • Avoid problematic parameter values, which may lead to erroneous predictions • Present proof-of-concept case study, reducing the parameter uncertainty by at least 10.3924909 % Semi-empirical models for LLE are essential for chemical process design but rely on experimental data for parameter estimation. In the face of high experimental costs, determining an informative experimental plan is crucial for constructing and validating these models efficiently. OED for (bounded-error) guaranteed GPE is a valuable approach to planning LLE experiments, but it has been limited to systems with simple input-output relations. We extend this method to systems whose input-output relation is implicitly given by an equation system with additional semi-infinite constraints, which corresponds to the rigorous computation of an LLE. Excess Gibbs free energy models are highly flexible but can predict spurious phase splits due to (i) using non-rigorous computations, or (ii) parameter values that are problematic in the sense that they lead to erroneous model behavior. To mitigate these issues in experiment planning, we (i) employ rigorous calculations using Baker’s criterion, and (ii) enforce additional requirements, based on [Mitsos et al. Chem. Eng. Sci., 64(3):548–559, 2009] to exclude parameter values that lead to erroneous model behavior. We solve the resulting problem using (generalized) semi-infinite programming techniques and provide a proof of concept using the Redlich-Kister model to plan the OED to reduce parameter uncertainty. Our approach successfully avoids wrong predictions due to problematic parameter values and computes an experimental design that significantly reduces the predicted parameter uncertainty. However, our method is computationally demanding, necessitating advancements in numerical methods for practical applications involving multiple parameters or more complex excess models.

This paper addresses the problem of interval estimation of states and faults for local nonlinear Takagi-Sugeno fuzzy systems. A novel estimation method integrating observer design and zonotopic analysis is proposed. Firstly, a new discrete-time intermediate observer is designed. By introducing two auxiliary variables, the design freedom of the observer is enhanced, leading to improved interval estimation performance. Secondly, to handle the local nonlinear dynamics within the system, a semi-infinite programming method based on zonotopic analysis is developed. This method enables the online computation of minimal outer-bounding zonotopes for the local nonlinear dynamics, achieving precise bounding of their effects. Furthermore, by incorporating the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{\infty }$</tex-math></inline-formula> and P-radius performance indices, a convex optimization condition is derived to ensure the stability of the error system and satisfy the required robustness performance. Finally, simulation results based on a nonlinear vehicle system are presented to demonstrate the effectiveness of the proposed approach.

Convex semi-infinite programming for station keeping in geostationary orbit

ABSTRACT This study presents a semi-infinite interval type-2 fuzzy multi-objective programming (SIIT2F-MOP) model for optimizing irrigation water management under uncertainty. The model integrates fuzzy sets, linear programming, and semi-infinite programming to address dynamic parameters. Applied to Zhangzhou City, Fujian, it optimizes water allocation for three crops across 11 districts and provides interval-based results. Combined with regional hydrological-characteristic methods, the model improves economic, yield, and water-saving benefits. At H3, α = 0.8, the results of the SIIT2F-MOP model under the three objectives of economic, crop yield, and water-saving are [686 930.9, 796 782.6] × 103 yuan, [394 613.7, 511 387.9] × 103 kg, and [7645.6, 148 567.6] × 103 m3, respectively. Compared with other methods, interval ranges decreased by up to 17.3%, 11.8%, and 8.6%, showing stronger performance under water-resource uncertainty. The analysis shows flexibility in assessing benefits and trade-offs under varied water-supply scenarios. The SIIT2F-MOP model supports evidence-based policy for sustainable agricultural and water-resource management.

Robust optimality and duality for multiobjective semi-infinite programming problems with equilibrium constraints under data uncertainty

Constraint qualifications and optimality criteria for robust conic multiobjective semi-infinite programming problems with vanishing constraints on Riemannian manifolds

In the present paper, we study the Lagrange dualities of the DC infinite optimization problems. By using the properties of the epigraph of the conjugate functions, we provide characterizations for the DC infinite optimization problem to have the stable weak Lagrange dualities and the stable Lagrange dualities. Applications to DC semi-infinite programming, DC conic programming and convex infinite programming are given.

This paper investigates the supplier’s pricing problem under upfront reservation discount (URD) contracts where the buyer reserves products in advance and then adjusts the purchase quantity based on realized end-market demand. A key challenge is that the supplier typically has limited data to estimate the demand distribution and possesses inferior information compared to the buyer. To address the challenges of distributional ambiguity and information asymmetry, we develop a refined distributionally robust optimization model for the supplier’s URD pricing to maximize her worst-case profit. To better infer true demand patterns, beyond the conventional reliance on historical demand data, our approach leverages past transaction records involving supplier–buyer interactions through the inverse optimization underlying the first-order conditions of the buyer’s newsvendor behavior. Then, a general Wasserstein p -distance minimization problem for p ≥ 1 is developed to generate a Refined Empirical Distribution (RED) in the enhanced set. We prove that the RED provides a superior estimation of the true demand distribution compared to the classical empirical distribution when the buyer holds an informational advantage. Although identifying the RED leads to an intractable semi-infinite program, we show that the RED admits a closed-form solution. To obtain the supplier’s worst-case profit involving a nonconvex distributionally optimistic optimization problem with a decision-dependent uncertainty set, we exploit the monotone transport structure between univariate distributions to truncate the distributions and convert the decision-dependent quantile constraints, which results in a finite-dimensional convex model that can be efficiently solved. Moreover, we extend the model to accommodate data noise, volatile market prices, evolving market conditions, and multi-item settings. Numerical experiments based on a virtual machine reservation problem in the cloud service market demonstrate the effectiveness and robustness of the proposed approach.

Approximate mixed type duality for semi-infinite programs having equilibrium constraints

Abstract Time-series information needs to be incorporated into energy system optimization to account for the uncertainty of renewable energy sources. Typically, time-series aggregation methods are used to reduce historical data to a few representative scenarios but they may neglect extreme scenarios, which disproportionally drive the costs in energy system design. We propose the robust energy system design (RESD) approach based on semi-infinite programming and use an adaptive discretization-based algorithm to identify worst-case scenarios during optimization. The RESD approach can guarantee robust designs for problems with nonconvex operational behavior, which current methods cannot achieve. The RESD approach is demonstrated by designing an energy supply system for the island of La Palma. To improve computational performance, principal component analysis is used to reduce the dimensionality of the uncertainty space. The robustness and costs of the approximated problem with significantly reduced dimensionality approximate the full-dimensional solution closely. Even with strong dimensionality reduction, the RESD approach is computationally intense and thus limited to small problems.

Asymptotic Analysis for a Class of Quasiconvex Semi-Infinite Programming Problems

A Global Approach for Generalized Semi-Infinite Programs with Polyhedral Parameter Sets

In this paper, we propose an approximation technique for a function which is the supremum of a general family of functions by means of a function, of LogExp-type, involving a finite number of the data functions.A study of variational properties of such an approximating function is carried out in the paper.In particular, the epigraphic convergence of these approximating functions to the supremum function is proven.Moreover, refined calculus specifying the relations among the subdifferentials (regular and general) of the approximating and the data functions are provided in the first part of the paper.In the second part, we propose to approximate a general semi-infinite programming problem by a simple problem with a single constraint.Applying the results of the first part we establish, under standard hypotheses, the convergence of optimal values and solutions of these problems to the corresponding ones of the original problem.The Lagrangian duality of this problem is also studied but restricted to the semi-infinite convex programming problem, and optimal dual solutions are built by means of a sequential procedure.

Bounding-focused discretization methods for the global optimization of nonconvex semi-infinite programs

Abstract Existing algorithms for generalized semi-infinite programs can only handle lower-level constraints containing equality constraints depending on upper-level variables (so-called coupling equality constraints) under limiting assumptions. More specifically, discretization-based algorithms require that the coupling equality constraints result in some lower-level variables being determined uniquely as implicit functions of the other lower-level and upper-level variables. We propose an adaptation of the discretization-based algorithm of Blankenship &amp; Falk and demonstrate it can handle coupling equality constraints under the weaker assumption of stability of the solution set for these constraints in the sense of Lipschitz lower semi-continuity. The key idea is to allow a perturbation of the lower-level variable values from discretization points in connection with changes in the upper-level variables in the discretized upper-level problem. We enforce that these perturbed values satisfy the coupling equality constraints while remaining close to the discretization point, provided we can guarantee the stability of the solution in the sense that a nearby solution exists for small changes of the upper-level variables. We provide concrete realizations of the algorithm for three different situations: i) when knowledge about a certain Lipschitz constant is available, ii) when the coupling equality constraints are assumed to have full rank, and iii) when the coupling equality constraints are additionally linear in the lower-level variables. Numerical experiments on small test problems and a physically motivated problem related to power flow illustrate that the approach can be successfully applied to solve the challenging problems, but is currently limited in terms of scalability.

In this paper, we formulate generalized (C,α,η,ρ,d)-invexity and based on these definitions, we derive several sufficient conditions for optimality in multiobjective semi-infinite programming problems. Further, under the assumptions of dual model we solve corresponding weak, strong and strict converse duality theorems for these multiobjective semiinfinite programming problem.

Abstract This paper studies generalized semi-infinite programs (GSIPs) given by polynomials. We propose a hierarchy of polynomial optimization relaxations to solve them. They are based on Lagrange multiplier expressions and polynomial extensions. Moment-SOS relaxations are applied to solve the polynomial optimization. The convergence of this hierarchy is shown under certain conditions. In particular, the classical semi-infinite programs can be solved as a special case of GSIPs. We also study GSIPs that have convex infinity constraints and show that they can be solved exactly by a single polynomial optimization relaxation. The computational efficiency is demonstrated by extensive numerical results.

This paper introduces a refined Wasserstein distributionally robust optimization (RWDRO) model to address contract pricing under information asymmetry. Our RWDRO model improves on traditional Wasserstein DRO (WDRO) models that rely solely on pure demand data by refining the Wasserstein ambiguity set through inverse optimization techniques applied to the buyer’s historical order data. To address the computational challenges arising from semi-infinite programming in determining the new center distribution of the Wasserstein ball with dual-source data, we propose an equivalent linear programming approach leveraging Lagrange duality and set partitioning techniques. Then, we establish bounds for the buyer’s worst-case order quantity and the seller’s worst-case profit using first-order conditions. For dependent multiproduct cases, we propose a partition-based cutting-plane algorithm to obtain an σ-optimal solution. For single-product and independent multiproduct cases, we develop a tractable second-order cone programming model. Numerical experiments highlight the superior out-of-sample performance of RWDRO over traditional WDRO models, especially in small-data regimes, and the computational efficiency of our proposed solution methods. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods &amp; Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72271145, 72134004, and 92367202]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0547 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0547 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

$$\mathcal {E}$$-LU-pareto solutions for semi-infinite programs with multiple intervals

Why engineers should care about semi-infinite programming: Nominal versus tolerance-aware geometry optimization of a proportional electromagnetic actuator