Sufficient Optimality Conditions Research Articles

An Efficient Global Optimality Certificate for Landmark-Based SLAM

Modern state estimation is often formulated as an optimization problem and solved using efficient local search methods. These methods at best guarantee convergence to local minima, but, in some cases, global optimality can also be certified. Although such global optimality certificates have been well established for 3D <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pose-graph optimization</i> , the details have yet to be worked out for the 3D landmark-based SLAM problem, in which estimated states include both robot poses and map landmarks. In this letter, we address this gap by using a graph-theoretic approach to cast the subproblems of landmark-based SLAM into a form that yields a sufficient condition for global optimality. Efficient methods of computing the optimality certificates for these subproblems exist, but first require the construction of a large data matrix. We show that this matrix can be constructed with complexity that remains linear in the number of landmarks and does not exceed the state-of-the-art computational complexity of one local solver iteration. We demonstrate the efficacy of the certificate on simulated and real-world landmark-based SLAM problems. We also integrate our method into the state-of-the-art SE-Sync pipeline to efficiently solve landmark-based SLAM problems to global optimality. Finally, we study the robustness of the global optimality certificate to measurement noise, taking into consideration the effect of the underlying measurement graph.

Learning of neural network with optimal control tools

Any learned artificial neural network on a given set of observations represents a function of several variables with vector values or real values. However, in general it is unknown except very simply cases and we have trouble to tell anything about its properties behind very general results received from learning data. In applications, such as medicine, it needs to say not only that on training data we get some error, we have to know that an error is not greater than some ɛ for all data for which we consider the system.It is well known that the learned neural network define a function. However, we check correctness of it only on finite number of observed data. We develop an optimal control approach allowing to find approximation of an unknown function realizing the given observable data, parametrized by a set of controls and defined as ordinary differential equations. Moreover, to measure discrepancy, of the output of the network we define a functional into which we include probability distribution function estimating distribution of the data.We develop a dual dynamic programming ideas to formulate a new optimization problem. We apply it to derive and to prove sufficient approximate optimality conditions for approximate neural network which should work correctly for given ɛ with respect to built functional, on a data different than the set of observations.

Second-order optimality conditions in nonsmooth vector optimization

Abstract In this paper, we introduce new classes of nonsmooth second-order cone-convex functions and respective generalizations in terms of first and second-order directional derivative. These classes encapsulate several already existing classes of cone-convex functions and their weaker variants. Second-order KKT type sufficient optimality conditions and duality results for a nonsmooth vector optimization problem are proved using these functions. The results have been supported by examples.

Identifying approximate proper efficiency in an infinite dimensional space

The main idea of this article is to characterize approximate proper efficiency that is a widely used optimality concept in multicriteria optimization problems that prevents solutions having unbounded trade-offs. We analyze a modification of approximate proper efficiency for problems with infinitely many objective functions. We obtain some necessary and sufficient optimality conditions for this modification of approximate proper efficiency. This modified version of approximation guarantees the general characterizations of approximate properly efficient points as solutions to weighted sum problems and modified weighted Tchebycheff norm problems, even if there is an infinite number of criteria. The provided proofs concerning the modified definition show that if the number of the objective functions is infinite, then these results become invalid under the primary definition of approximate proper efficiency.

Local Optimality Conditions for a Family of Hidden Convex Optimization

Hidden convex optimization is a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we study a family of hidden convex optimization that joints the classical trust region subproblem (TRS) with convex optimization (CO). It also includes p-regularized subproblem (p &gt; 2) as a special case. We present a comprehensive study on local optimality conditions. In particular, a sufficient condition is given to ensure that there is at most one local nonglobal minimizer, and at this point, the standard second-order sufficient optimality condition is necessary. To our surprise, although (TRS) has at most one local nonglobal minimizer and (CO) has no local nonglobal minimizer, their joint problem could have any finite number of local nonglobal minimizers. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12171021, 12131004, and 11822103], the Beijing Natural Science Foundation [Grant Z180005], and the Fundamental Research Funds for the Central Universities. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2023.0089 .

The Necessary and Sufficient Optimality Conditions for a System of FOCPs with Caputo–Katugampola Derivatives

The necessary optimality conditions with Lagrange multipliers are studied and derived for a new class that includes the system of Caputo–Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo–Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time and the final state are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.

New approach to fixed-time stability of a nonlinear system

The problem of optimal fixed-time controllability to zero for an ordinary differential equation is investigated. We were inspired by the papers Lopez-Ramirez et al. (2018) and Polyakov (2012). We assume that our system is finite time stable. The initial conditions of trajectories we treat as control. We want to maximize the time when the trajectory starting at initial condition terminates at zero. We continue the control approach to stability from Polyakov (2012) to construct sufficient optimality conditions for the fixed-time stability in terms of the optimal control theory and find closed-loop control (feedback control) to steer the trajectory to zero.

A new scalarization approach and applications in set optimization

In this paper, we introduce a new nonlinear scalarization function and apply it to investigate optimality conditions and scalar representations for minimal sets and set optimization problems with the set less order relation. First, we propose a new scalarization function for sets belonging to the power set of a normed space and discuss its properties. Next, we apply such properties to formulate sufficient and necessary optimality conditions for strictly minimal and weakly minimal sets. Finally, under suitable adjustments, we provide nonlinear scalarization functions and investigate scalar representations of strictly efficient solution and weakly efficient solution sets of the reference problems.

Second-order optimality conditions for locally Lipschitz vector optimization problems

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives.

Focus on sufficient optimality conditions in D.C. vector optimization

Sufficient optimality conditions for a D.C. vector optimization problem are given by Guo and Li (Journal of Optimization Theory and Applications 162, 821–844, 2014). Guo and Li's main result turned out to be false and was corrected by Shafie and Bozorgnia (Journal of Optimization Theory and Applications 182, 837–849, 2019). In this work, we have pointed out that Shafie and Bozorgnia's result is wrong. We correct the flaws and present an alternative proof for the main result.

Robust domination in random graphs

In this paper, we study “robust” dominating sets of random graphs that retain the domination property even if a small deterministic set of edges are removed. We motivate our study by illustrating with examples from wireless networks in harsh environments. We then use the probabilistic method and martingale difference techniques to determine sufficient conditions for the asymptotic optimality of the robust domination number. We also discuss robust domination in sparse random graphs where the number of edges grows at most linearly in the number of vertices.

Semi-infinite Mathematical Programming Problems involving Generalized Convexity

In this paper, we consider semi-infinite mathematical programming problems withequilibrium constraints (SIMPEC). By using the notion of convexificators, we establish sufficient optimality conditions for the SIMPEC. We formulate Wolfe and Mond-Weir type dual models for the SIMPEC under generalized convexity assumptions. Moreover, weak and strong duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators.

Optimal control of higher-order retarded differential inclusions

The article investigates an optimal control problem described by retarded differential inclusions (DFIs) of higher order with endpoint constraints. In terms of the Euler-Lagrange type adjoint inclusions and the Hamiltonian, a sufficient optimality condition is derived for higher-order DFIs that have different forms in different time intervals depending on the delay parameter, which in the problem without delay, these two adjoint inclusions coincide. It is shown that the adjoint inclusion for the DFIs of the first order, defined in terms of locally conjugate mappings, coincides with the classical Euler-Lagrange inclusion. Thus, the obtained results are universal in the sense that sufficient optimality conditions can be formulated for a DFI of any order. At the end of the paper, problems with a high-order polyhedral DFIs and higher-order linear optimal control problems are considered, the optimality conditions of which are transformed into the Pontryagin maximum principle. Also, for high-order polyhedral optimization with delay, from the point of view of abstract economics, non-negative adjoint variables can be interpreted as the price of a resource.

Условие оптимальности типа принципа максимума Понтрягина в задаче управления линейными разностными уравнениями дробного порядка

The optimal control problem of fractional order linear two-dimensional difference equations sys-tems is considered. It is assumed that the control function is included in the boundary condition, and the functional is linear. A necessary and sufficient optimality condition is proved in the discrete maximum principal form. A sufficient optimality condition is proved in the nonlinear but convex cost functional case.

SOME PROPERTIES OF {k}-PACKING FUNCTION PROBLEM IN GRAPHS

The recently introduced {k}-packing function problem is considered in this paper. Relationship between cases when k = 1, k ≥ 2 and linear programming relaxation are introduced with sufficient conditions for optimality. For arbitrary simple connected graph

Local minimizers for variational obstacle avoidance on Riemannian manifolds

&lt;abstract&gt;&lt;p&gt;This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories—called $ Q $-local minimizers and $ \Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.&lt;/p&gt;&lt;/abstract&gt;

МОДЕЛЮВАННЯ ОПТИМАЛЬНИХ ЗВ’ЯЗКІВ МІЖ ТІНЬОВОЮ ТА ЛЕГАЛЬНОЮ ЕКОНОМІКАМИ

This article is devoted to modelling the optimal links between the shadow and legal economies in order to understand their interaction and impact on the main patterns of changes in the savings of employees and business owners, and in the price of aggregate goods. The study focuses on finding optimal management strategies that will reduce the size of the shadow economy and improve the economic situation. The article uses the theory of optimal control to build a mathematical model of the interaction between the shadow and legal economies. A number of assumptions are taken into account for construction the mathematical model, in particular, firstly, the rate of change of savings of real sector employees is proportional to the difference between their total income, consisting of legal and shadow salaries, reduced by income taxes, and total expenditures (expenditures on consumption of legal and shadow products); secondly, the dynamics of owners' savings depends on the difference between their total income from the sale of legal and shadow products, reduced by the amount of relevant income taxes, and total expenditures on employee salaries, production needs and taxation in the legal and shadow sectors and thirdly, the rate of change in the price of legal and shadow products is proportional to the difference between total demand and total supply of these products on the market. The assumptions are added with initial conditions for the dynamics of cash savings and the price of the aggregate product, and restrictions are imposed on the consumer shares of savings used for consumption of legal and shadow products for employees and for business owners. The goal criterion is to maximise the average (integral) total welfare of employees and entrepreneurs over a certain period of time. Sufficient conditions of optimality were used for the study. The findings may be useful for economists, policy makers and researchers interested in the shadow economy and looking for effective ways to influence this sector. The results of the study can serve as a basis for the development of policies and recommendations aimed at reducing the shadow economy and stimulating legal economic activity.

ТЕХНОЛОГІЯ ВИКОРИСТАННЯ ТЕОРІЇ ОПТИМАЛЬНОГО КЕРУВАННЯ ДЛЯ МОДЕЛЮВАННЯ ОДНОПРОДУКТОВОЇ МАКРОЕКОНОМІКИ ЗРОСТАННЯ

Research in macroeconomic growth remains extremely relevant and important in the modern world. Macroeconomic growth focuses on analysing the dynamics of production, income, and consumption, enabling the development of scenarios for long-term development. The technology of using the theory of optimal control for modelling single-product growth macroeconomics plays an important role in understanding and forecasting economic development. The article proposes a technology for using the theory of optimal control to model a single-product growth macroeconomy. Technological tasks for modelling single-product macroeconomic growth are formulated, which can be initial (at the beginning of the production process), intermediate throughout the entire duration of the process, and final at the end state of production. In order to construct a model of optimal macroeconomic growth, several assumptions are taken into account. Specifically, gross output is decomposed into final output and production consumption. Production consumption is assumed to be proportional to gross output. Final output is further divided into non-production consumption, total investment, government expenditure, taxation, pollution abatement, and balance of trade (exports minus imports). To build the optimal process, where consumption rate acts as the control variable and specific capital represents the phase trajectory, a target function (goal criterion) is formalized. The target function aims to maximize the average (integral) discounted consumption over a specified time interval. Sufficient optimality conditions are used to investigate the model of optimal control. The Lagrange method is employed to account for the constraint on minimum consumption. This allows transforming the constrained optimization problem into an unconstrained optimization problem involving the optimization of two functions with multiple variables. During the investigation of the single-product macroeconomic growth model, an algorithm has been developed to calculate the optimal process for three formulated technological tasks under selected three-tier regimes. Among the built optimal processes, it is possible to select a priority optimal process.

Necessary and sufficient optimality conditions for fractional Fornasini-Marchesini model

In the paper, we study an optimal control problem connected with a fractional Fornasini-Marchesini model described by partial Caputo derivatives. Using a new version of the increment method, the necessary and sufficient optimality condition is obtained in the form of the Pontryagin maximum principle for the posed fractional optimal control problem. The paper is concluded with illustrating example.

Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem

&lt;p style='text-indent:20px;'&gt;In this work, we are concerned with a fractional multiobjective optimization problem &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ (P) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; involving set-valued maps. Based on necessary optimality conditions given by Gadhi et al. [&lt;xref ref-type="bibr" rid="b14"&gt;14&lt;/xref&gt;], using support functions, we derive sufficient optimality conditions for &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \left( P\right) , $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and we establish various duality results by associating the given problem with its Mond-Weir dual problem &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \left( D\right) . $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; The main tools we exploit are convexificators and generalized convexities. Examples that illustrates our findings are also given.&lt;/p&gt;