Infinite-dimensional Optimization Problem Research Articles

A combined work optimization technology under resource constraints with an application to road repair

We propose an approach for solving the task prioritization problem in road surface repair under bounded resources; the idea is to use a combination of defect recognition and classification methods based on statistical analysis and machine learning (random forests) with original methods for solving infinite-dimensional optimization problems (optical-geometric analogy). We show the results of a computational experiment that indicate high performance of the developed algorithms, and the resulting solutions were evaluated highly by experts in road facilities management. Our results may encourage more efficient use of resources to improve the quality of motorways.

Pareto optimality and existence of quasi-equilibrium in exchange economies with an indefinite future

We study the attainability of Pareto optimal allocations and existence of quasi-equilibrium in exchange economies where agents have utility functions that value consumption in an indefinite future. These utility functions allow for fairly general discounting of consumption over finite time horizons, but add a utility weight to the bulk of the consumption sequence, which we identify with the indefinite future. As our commodity space we use the space of all convergent sequences with the limit of the sequence representing consumption in the indefinite future. We derive a necessary and sufficient condition for the attainability of the Pareto optimal allocations. This condition implies that efficiency can only be attained if consumers’ valuations of time are very similar. Our proof relies on the existence of an interior solution to certain infinite dimensional optimization problems. If the condition is not met, no interior quasi-equilibria exist. We extend the model to include consumers with Rawlsian-like maximin utility.

Solving infinite horizon optimization problems through analysis of a one-dimensional global optimization problem

Infinite horizon optimization (IHO) problems present a number of challenges for their solution, most notably, the inclusion of an infinite data set. This hurdle is often circumvented by approximating its solution by solving increasingly longer finite horizon truncations of the original infinite horizon problem. In this paper, we adopt a novel transformation that reduces the infinite dimensional IHO problem into an equivalent one dimensional optimization problem, i.e., minimizing a Holder continuous objective function with known parameters over a closed and bounded interval of the real line. We exploit the characteristics of the transformed problem in one dimension and introduce an algorithm with a graphical implementation for solving the underlying infinite dimensional optimization problem.

The Slater Conundrum: Duality and Pricing in Infinite-Dimensional Optimization

Duality theory is pervasive in finite-dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented by a vector of “dual prices” that index the primal constraints and have a natural economic interpretation. In infinite dimensions, we show that this simple dual structure, and its associated economic interpretation, may fail to hold for a broad class of problems with constraint vector spaces that are Riesz spaces (ordered vector spaces with a lattice structure) that either are $\sigma$-order complete or satisfy the projection property. In these spaces we show that the existence of interior points required by common constraint qualifications for zero duality gap (such as Slater's condition) implies the existence of ...

On the Curse of Dimensionality in the Ritz Method

It is shown that the classical Ritz method of the calculus of variations suffers from the "curse of dimensionality," i.e., an exponential growth, as a function of the number of variables, of the dimension a linear subspace needs in order to achieve a desired relative improvement in the accuracy of approximation of the optimal solution value. The proof is constructive and is obtained by exhibiting a family of infinite-dimensional optimization problems for which this happens, namely those with quadratic functional and spherical constraint. The results provide a theoretical motivation for the search of alternative solution methods, such as the so-called "extended Ritz method," to deal with the curse of dimensionality.

Lagrange Duality for Evenly Convex Optimization Problems

An evenly convex function on a locally convex space is an extended real-valued function, whose epigraph is the intersection of a family of open halfspaces. In this paper, we consider an infinite-dimensional optimization problem, for which both objective function and constraints are evenly convex, and we recover the classical Lagrange dual problem for it, via perturbational approach. The aim of the paper was to establish regularity conditions for strong duality between both problems, formulated in terms of even convexity.

On the use of the quasi-relative interior in optimization

The notion of quasi-relative interior was introduced by Borwein and Lewis in 1992 and applied for duality results in partially finite convex optimization problems. In the last 10 years, several articles were dedicated to duality results in infinite-dimensional scalar, vector and set-valued optimization problems using this notion. The aim of this paper is to refine and discuss such results. We do this observing that the notion of quasi-relative interior is related to (non-proper) separation of a convex set and some of its elements, then pointing out the relation between the subdifferentiability of a function associated to a set of epigraph type at a certain point and the fact that a corresponding point is not in the quasi-relative interior of the closed convex hull of the set.

Shape deformation analysis from the optimal control viewpoint

A crucial problem in shape deformation analysis is to determine a deformation of a given shape into another one, which is optimal for a certain cost. It has a number of applications in particular in medical imaging.In this article we provide a new general approach to shape deformation analysis, within the framework of optimal control theory, in which a deformation is represented as the flow of diffeomorphisms generated by time-dependent vector fields. Using reproducing kernel Hilbert spaces of vector fields, the general shape deformation analysis problem is specified as an infinite-dimensional optimal control problem with state and control constraints. In this problem, the states are diffeomorphisms and the controls are vector fields, both of them being subject to some constraints. The functional to be minimized is the sum of a first term defined as geometric norm of the control (kinetic energy of the deformation) and of a data-attachment term providing a geometric distance to the target shape.This point of view has several advantages. First, it allows one to model general constrained shape analysis problems, which opens new issues in this field. Second, using an extension of the Pontryagin maximum principle, one can characterize the optimal solutions of the shape deformation problem in a very general way as the solutions of constrained geodesic equations. Finally, recasting general algorithms of optimal control into shape analysis yields new efficient numerical methods in shape deformation analysis. Overall, the optimal control point of view unifies and generalizes different theoretical and numerical approaches to shape deformation problems, and also allows us to design new approaches.The optimal control problems that result from this construction are infinite dimensional and involve some constraints, and thus are nonstandard. In this article we also provide a rigorous and complete analysis of the infinite-dimensional shape space problem with constraints and of its finite-dimensional approximations.

Co-design of event-triggered [formula omitted] control for discrete-time linear parameter-varying systems with network-induced delays

The additional threshold of a mixed event-triggering mechanism makes it impossible to guarantee the requirement of asymptotic stability. As a consequence, it is necessary to investigate uniform ultimate boundedness which is a mild stability property. In this paper, we address event-triggered H∞ control for discrete-time linear parameter-varying systems by jointly designing a mixed event-triggering mechanism and a set of state feedback controllers. In the frame of time-delay system, we propose a parameter-dependent sufficient condition such that the closed-loop system with mixed event-triggering mechanism is globally uniformly ultimately bounded with a minimized ultimate bound for a certain disturbance attenuation level. As an extensive study, such sufficient condition is proved to be in a sense of global asymptotic stability when the additional threshold tends to zero. To avoid the infinite dimensional optimization problem, the proposed parameter-dependent co-design condition is relaxed as a finite set of linear matrix inequalities at the polytope vertices, which can be numerically trackable and computationally efficient. Two numerical examples are provided to demonstrate the effectiveness of the proposed method.

Safety Analysis of Stochastic Dynamical Systems

This paper presents a method for verifying the safety of a stochastic system. In particular, we show how to compute the largest set of initial conditions such that a given stochastic system is safe with probability p.To compute the set of initial conditions we rely on the moment method that via Haviland's theorem allows an infinite dimensional optimization problem on measures to be formulated as a polynomial optimization problem. Subsequently, the moment sequence is truncated (relaxed) to obtain a finite dimensional polynomial optimization problem. Finally, we provide an illustrative example that shows how the p-safe initial set is computed numerically.

METHOD COMPARISON FOR ESTIMATION OF DISTRIBUTED PARAMETERS IN PERMITTIVITY MODELS USING REFLECTANCE

The use of reflectance spectroscopy is a current area of study for the non-invasive evaluation of complex materials such as ceramic matrix composites. In order to model the reflectance, one must specify a model for the complex permittivity. In this work we compare two methods for modeling the complex permittivity of a heterogenous material. In one approach, we impose a probability distribution on a subset of the dielectric parameters. This approach leads to an infinite dimensional optimization problem over the space of probability measures. We approximate this space with a finite dimensional space by using either a Dirac approximation method or a linear spline approximation method. The second approach is to assume a number of oscillators in the permittivity model, and then use a convolution with a normal distribution. We compare both of these approaches on simulated data sets as well as data obtained from inorganic glasses. Each of these methods are able to fit the data well, yet the ease in interpreting the estimation results of imposing a probability distribution on parameters, as well as the tight mathematical results [2, 7] guaranteeing convergence under the Prohorov metric, lead us to favor the first approach.

The Infinite-Dimensional Optimal Filtering Problem with Mobile and Stationary Sensor Networks

In this article, we introduce a framework to address filtering and smoothing with mobile sensor networks for distributed parameter systems. The main problem is formulated as the minimization of a functional involving the trace of the solution of a Riccati integral equation with constraints given by the trajectory of the sensor network. We prove existence and develop approximation of the solution to the Riccati equation in certain trace-class spaces. We also consider the corresponding optimization problem. Finally, we employ a Galerkin approximation scheme and implement a descent algorithm to compute optimal trajectories of the sensor network. Numerical examples are given for both stationary and moving sensor networks.

Robustifying Convex Risk Measures for Linear Portfolios: A Nonparametric Approach

This paper introduces a framework for robustifying convex, law invariant risk measures. The robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure, which represent the investors' beliefs about the distribution of future asset losses. It is shown that under mild conditions, the infinite dimensional optimization problem of finding the worst-case risk can be solved analytically and closed-form expressions for the robust risk measures are obtained. Using these results, robust versions of several risk measures including the standard deviation, the Conditional Value-at-Risk, and the general class of distortion functionals are derived. The resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems, and a numerical study shows that in most cases they perform significantly better out-of-sample than their nonrobust variants in terms of risk, expected losses, and turnover.

Suboptimality conditions for mathematical programs with equilibrium constraints

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form \vspace{-0.5cm} \begin{eqnarray*} 0\in G(x,y)+Q(x,y), \end{eqnarray*} \vspace{-0.6cm} \noindent where both mappings $G$ and $Q$ are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new {\em weak} and {\em strong suboptimality} conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do {\em not assume the existence of optimal solutions}. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techniques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental {\em extremal principle} in variational analysis and its analytic counterpart known as the {\em subdifferential variational principle.}

Finding points of maximal loadability considering post-contingency corrective controls

Lately, much work in the area of voltage stability assessment has been focused on finding post-contingency corrective controls. In this article a contribution to this area will be presented where we search for maximal loadability while considering post-contingency corrective controls. This objective is different from the usual approach to the problem, where the aim is to include the post-contingency controls in a security-constrained optimal power flow.Our approach gives us an optimal control problem with a variable start point. Optimal control problems are generally very cumbersome to solve in high dimensions. However, under some mild assumptions we find that our infinite dimensional optimization problem can be transformed into a finite dimensional one. More specifically, by assuming that the load recovery is an explicit function of time we can specify a set of constraints that are necessary for optimality.

Multiphase Optimal Control Framework for Commercial Aircraft Four-Dimensional Flight-Planning Problems

The calculation of a flight plan involves the consideration of multiple elements. They can be classified as either continuous or discrete, and they can include nonlinear aircraft performance, atmospheric conditions, wind forecasts, airspace structure, amount of departure fuel, and operational constraints. Moreover, multiple differently characterized flight phases must be considered in flight planning. The flight-planning problem can be regarded as a trajectory optimization problem. A natural way to address a trajectory optimization problem is using optimal control techniques. The multiphase nature of the problem and the nonlinear dynamics of the aircraft lead to the formulation of a multiphase optimal control problem. A Hermite–Simpson collocation method is employed to transcribe the infinite-dimensional optimal control problem into a finite-dimensional optimization one, which is solved using a nonlinear programming solver. An application to the optimal takeoff weight, minimum fuel consumption trajectory planning problem is presented. Aircraft performances, flight procedures, and the resulting fuel consumption are analyzed. The problem is also solved for different takeoff weights and different cost indices. Their effects in the optimal performances are presented and discussed.

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

Standard second order sufficient conditions in optimal control theory provide not only the information that an extremum is a weak local minimizer, but also tell us that the extremum is locally unique. It follows that such conditions will never cover problems in which the extremum is continuously embedded in a family of constant cost extrema. Such problems arise in periodic control, when the cost is invariant under time translations, in shape optimization, where the cost is invariant under Euclidean transformations (translations and rotations of the extremal shape), and other areas where the domain of the optimization problem does not really comprise elements in a linear space, but rather an equivalence class of such elements. We supply a set of sufficient conditions for minimizers that are not locally unique, tailored to problems of this nature. The sufficient conditions are in the spirit of earlier conditions for ‘non-isolated’ minima, in the context of general infinite dimensional nonlinear programming problems provided by Bonnans, Ioffe and Shapiro, and require coercivity of the second variation in directions orthogonal to the constant cost set. The emphasis in this paper is on the derivation of directly verifiable sufficient conditions for a narrower class of infinite dimensional optimization problems of special interest. The role of the conditions in providing easy-to-use tests of local optimality of a non-isolated minimum, obtained by numerical methods, is illustrated by an example in optimal control.

Empirical Approach for Optimal Reinsurance Design

This article proposes a novel and practical approach of addressing optimal reinsurance via an empirical approach. This method formulates reinsurance models using the observed data directly and has advantages including (1) transformation of an infinite dimensional optimization problem to a finite dimension, (2) no required explicit distributional assumption on the underlying risk, and (3) many empirical-based reinsurance models can be solved efficiently using the second-order conic programming. This allows insurers to incorporate many desirable objective functions and constraints while still retaining the ease of obtaining optimal reinsurance strategies. Numerical examples, including applications to actual Danish fire loss data, are provided to highlight the efficiency and the practicality of the proposed empirical models. The stability and consistency of the empirical-based solutions are also analyzed numerically.

Optimal control processes associated with a class of discontinuous control systems: applications to sliding mode dynamics

This paper presents a theoretical approach to optimal control problems (OCPs) governed by a class of control systems with discontinuous right-hand sides. A possible application of the framework developed in this paper is constituted by the conventional sliding mode dynamic processes. The general theory of constrained OCPs is used as an analytic background for designing numerically tractable schemes and computational methods for their solutions. The proposed analytic method guarantees consistency of the resulting approximations related to the original infinite-dimensional optimization problem and leads to specific implementable algorithms.

Sharp Inequalities for $f$-Divergences

f-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics, and information theory such as Kullback-Leibler divergence, chi-squared divergence, squared Hellinger distance, total variation distance, and so on. In this paper, we study the problem of maximizing or minimizing an f-divergence between two probability measures subject to a finite number of constraints on other f-divergences. We show that these infinite-dimensional optimization problems can all be reduced to optimization problems over small finite dimensional spaces which are tractable. Our results lead to a comprehensive and unified treatment of the problem of obtaining sharp inequalities between f-divergences. We demonstrate that many of the existing results on inequalities between f-divergences can be obtained as special cases of our results. We also improve on some existing non-sharp inequalities.