Agnieszka B. Malinowska, Delfim F. M. Torres
We study a multiobjective variational problem on time scales. For this problem, necessary and sufficient conditions for weak local Pareto optimality are given. We also prove a necessary optimality condition for the isoperimetric problem with multiple constraints on time scales.
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In this paper, we present some new necessary and sufficient optimality conditions in terms of the Clarke subdifferentials for approximate Pareto solutions of a nonsmooth vector optimization problem which has an infinite number of constraints. As a consequence, we obtain optimality conditions for the particular cases of cone-constrained convex vector optimization problems and semidefinite vector optimization problems. Examples are given to illustrate the obtained results.
We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations concerning the objective mapping and the constraints are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation and the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pare…
One of the most important optimality conditions to aid to solve a vector optimization problem is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker condition. However, to obtain the sufficient optimality conditions, it is necessary to impose additional assumptions on the objective functions and in the constraint set. The present work is concerned with the constrained vector quadratic fractional optimization problem. It shows that sufficient Pareto optimality conditions and the main duality theorems can be established without the assumption of generalized con…
We formalize the concept of Pareto Adaptive Robust Optimality (PARO) for linear Adaptive Robust Optimization (ARO) problems. A worst-case optimal solution pair of here-and-now decisions and wait-and-see decisions is PARO if it cannot be Pareto dominated by another solution, i.e., there does not exist another such pair that performs at least as good in all scenarios in the uncertainty set and strictly better in at least one scenario. We argue that, unlike PARO, extant solution approaches -- including those that adopt Pareto Robust Optimality from static robust optimization -- could fail in ARO …
Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body $\mathfrak x$, we try to maximize the volume of $\mathfrak x$ and minimize the width of $\mathfrak x$ simultaneously. These problems are addressed along the lines of multiple criteria decision making.
There are several concepts and definitions that characterize and give optimality conditions for solutions of a vector optimization problem. One of the most important is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker condition. This condition ensures the existence of an arbitrary neighborhood that contains an local optimal solution. The present work we introduce an alternative concept to identify the local optimal solution neighborhood in vector optimization problems. The main aspect of this contribution is the development of necessary and sufficient Pare…
Topological sorting is an important technique in numerous practical applications, such as information retrieval, recommender systems, optimization, etc. In this paper, we introduce a problem of generalized topological sorting with maximization of choice, that is, of choosing a subset of items of a predefined size that contains the maximum number of equally preferable options (items) with respect to a dominance relation. We formulate this problem in a very abstract form and prove that sorting by k-Pareto optimality yields a valid solution. Next, we show that the proposed theory can be useful in…
Pareto-Optimal Linear Programming
本文提出受帕累托最优性条件约束的线性规划问题Max-Pareto，证明其为NP完全，并给出双线性规划公式以求解最高福利的帕累托最优分配。
This paper deals with Pareto solutions of a nonsmooth fractional interval-valued multiobjective optimization. We first introduce four types of Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these solutions. Sufficient conditions for Pareto solutions of such a problem are also provided by means of introducing the concepts of (strictly) generalized convex functions defined in terms of the limiting/Mordukhovi…
Cardinality-Constrained Multi-Objective Optimization: Novel Optimality Conditions and Algorithms
针对带稀疏性约束的多目标优化问题，提出L-平稳性概念并分析其与现有最优性条件的关系，设计两种收敛性有理论保障的新算法，数值实验验证了方法有效性。
We model the formation of multi-layer transportation networks as a multi-objective optimization process, where service providers compete for passengers, and the creation of routes is determined by a multi-objective cost function encoding a trade-off between efficiency and competition. The resulting model reproduces well real-world systems as diverse as airplane, train and bus networks, thus suggesting that such systems are indeed compatible with the proposed local optimization mechanisms. In the specific case of airline transportation systems, we show that the networks of routes operated by ea…
Second-Order Necessary Optimality Conditions for Multi-Objective Optimal Control Problems on Riemannian Manifolds
本文研究黎曼流形上常微分方程的多目标最优控制问题，推导固定及自由终端时间下弱帕累托最优解的二阶必要条件，发现其依赖曲率张量，并通过示例说明与现有结果的差异。
We consider a Ramsey model with several households with heterogeneous preferences who are able to borrow capital to each other. Since the capital constraints of one household then depends on the others' capital, one can no longer optimize each household's welfare individually. This problem formulation leads to a Pareto optimization problem. We consider existence and first order optimality conditions as well as some numerical results.
Muscle Coactivation in the Sky: Geometry and Pareto Optimality of Energy vs. Aerodynamic Promptness and Multirotors as Variable Stiffness Actuators
本文通过引入气动响应这一动态指标，研究多旋翼中能量消耗与气动响应的几何与帕累托最优性，揭示了协同与对抗执行模式对飞行性能的影响，并提出动态'飞行肌肉'控制方法。
A general framework for providing interval representations of Pareto optimal outcomes for large-scale bi- and tri-criteria MIP problems
本文针对大规模双目标和三目标混合整数规划问题，提出一种在时间限制下提供帕累托最优结果区间表示的通用框架，用于导航帕累托前沿，并通过多目标0-1背包实验验证。
In this paper, we introduce the second-order subdifferentials for functions which are Gâteaux differentiable on an open set and whose Gâteaux derivative mapping is locally Lipschitz. Based on properties of this kind of second-order subdifferentials and techniques of variational analysis, we derive second-order necessary conditions for weak Pareto efficient solutions of multiobjective programming problems with constraints.
In this paper, we study second-order necessary and sufficient optimality conditions of Karush--Kuhn--Tucker-type for locally optimal solutions in the sense of Pareto to a class of multi-objective optimal control problems with mixed pointwise constraint. To deal with the problems, we first derive second-order optimality conditions for abstract multi-objective optimal control problems which satisfy the Robinson constraint qualification. We then apply the obtained results to our concrete problems. The proofs of obtained results are direct, self-contained without using scalarization techniques.
Integral Global Optimality Conditions and an Algorithm for Multiobjective Problems
本文针对不一定可微的多目标问题，通过加权和与Chebyshev加权标量化方法将积分全局最优性条件从单目标扩展至多目标，并提出一种基于Chebyshev标量化的算法以逼近弱Pareto前沿，通过测试问题验证了其有效性。
In this paper we obtain second- and first-order optimality conditions of Kuhn-Tucker type and Fritz John one for weak efficiency in the vector problem with inequality constraints. In the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable. We introduce notions of KTSP-invex problem and second-order KTSP-invex one. We obtain that the vector problem is (second-order) KTSP-invex if and only if for every triple $(\bar x,\barλ,\barμ)$ with Lagrange multipliers $\barλ$ and $\barμ$ for the objective function and constraints respective…
This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approxima…
Strong Metric Subregularity of the optimality mapping and second-order sufficient optimality conditions in extremal problems with constraints
本综述总结了优化中强度量次正则性（SMSR）的近期成果，基于二阶充分最优性条件，为数学规划、变分法和最优控制三类约束优化问题的最优性映射提供了SMSR充分条件，确保数据小扰动下最优解与Lagrange乘子的微小变化。
On approximate Pareto solutions in nonsmooth interval-valued multiobjective optimization with data uncertainty in constraints
针对约束含数据不确定性的非光滑区间值多目标优化问题，定义多类近似Pareto解，推导其最优性条件与对偶关系，为含不确定约束的区间值多目标优化提供理论支撑。
Average-Cost MDPs with Infinite State and Action Sets: New Sufficient Conditions for Optimality Inequalities and Equations
本文针对具有Borel状态和动作集的平均成本无限时域MDPs，提出了转移概率弱连续和集值连续下最优性不等式与方程成立的新充分条件，并证明了确定性最优策略的存在性。
First and Second Order Necessary and Sufficient Optimality Conditions of Fritz John Type for Vector Problems over Cones
该研究利用Kakutani不动点定理和Hadamard方向导数，给出锥上向量问题Fritz John型一阶、二阶最优性必要条件的新证法，并提供了广义凸函数下弱全局有效性与局部有效性的充分条件。
New second order sufficient optimality conditions for state constrained parabolic control problems
针对具有逐点控制与状态约束的半线性抛物方程控制问题，本文推导了 $L^2(Q)$ 意义下的新二阶充分最优性条件，并将结果推广至三维及以下空间区域。
The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler-Lagrange type for the basic, isoperimetric, and Lagrange variational problems are proved, as well as transversality and sufficient optimality conditions. This allows to obtain necessary and sufficient Pareto optimality conditions for multiobjective fractional variational problems.
Second-order sufficient optimality conditions in the calculus of variations
该研究证明变分法中若干经典二阶充分最优性条件彼此等价，还提出一种易于应用的新型等价二阶条件——通过积分线性二阶初值问题并检验解在定义域内的正性即可判定最优性。
We suggest a novel approach for the efficient and reliable approximation of the Pareto front of sufficiently smooth unconstrained bi-criteria optimization problems. Optimality conditions formulated for weighted sum scalarizations of the problem yield a description of (parts of) the Pareto front as a parametric curve, parameterized by the scalarization parameter (i.e., the weight in the weighted sum scalarization). Its sensitivity w.r.t. parameter variations can be described by an ordinary differential equation (ODE). Starting from an arbitrary initial Pareto optimal solution, the Pareto front …
This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization probl…
This note establishes that a generalization of $c$-cyclical monotonicity from the Monge-Kantorovich problem with two marginals gives rise to a sufficient condition for optimality also in the multi-marginal version of that problem. To obtain the result, the cost function is assumed to be bounded by a sum of integrable functions. The proof rests on ideas from martingale transport.
In this note, we present an elementary proof for a well-known second-order sufficient optimality condition in nonlinear semidefinite optimization which does not rely on the enhanced theory of second-order tangents. Our approach builds on an explicit elementary computation of the so-called second subderivative of the indicator function associated with the semidefinite cone which recovers the best curvature term known in the literature.
Algorithms and Differential Game Representations for Exploring Nonconvex Pareto Fronts in High Dimensions
本文提出了一种基于微分博弈和HJ方程的算法，用于高维非凸帕累托前沿的探索，通过标量化方法和对偶算法有效缓解维度诅咒。
In this paper, we present some second-order sufficient conditions in terms of the Demyanov-Pevnyi's second-order directional derivatives for efficiency of $C^1$ vector optimization problems with constraints. Our results improve and generalize conditions obtained by various authors in recent papers.
We provide necessary and sufficient conditions for robust efficiency (in the sense of Ehrgott et al. (2014)) to multiobjective optimization problems that depend on uncertain parameters. These conditions state that a solution is robust efficient (under minimization) if it is optimal to a strongly increasing scalarizing function, and only if it is optimal to a strictly increasing scalarizing function. By counterexample, we show that the necessary condition cannot be strengthened to convex scalarizing functions, even for convex problems. We therefore define and characterize a subset of the robust…
In this paper we derive for a controlled stochastic evolution system on a Hilbert space sufficient conditions for optimality. Our result is derived by using its so-called adjoint backward stochastic evolution equation.
The case of time minimization for affine control systems with control on the disk is studied. After recalling the standard sufficient conditions for local optimality in the smooth case, the analysis focusses on the specific type of singularities encountered when the control is prescribed to the disk. Using a suitable stratification, the regularity of the flow is analyzed, which helps to devise verifiable sufficient conditions in terms of left and right limits of Jacobi fields at a switching point. Under the appropriate assumptions, piecewise regularity of the field of extremals is obtained.
Toward a multi-objective optimization robust problem, the variations in design variables and design environment pa-rameters include the small variations and the large varia-tions. The former have small effect on the performance func-tions and/or the constraints, and the latter refer to the ones that have large effect on the performance functions and/or the constraints. The robustness of performance functions is discussed in this paper. A post-optimality sensitivity analysis technique for multi-objective robust optimization problems is discussed and two robustness indices are introduced. The fi…
We present a unified study of first and second order necessary and sufficient optimality conditions for minimax and Chebyshev optimisation problems with cone constraints. First order optimality conditions for such problems can be formulated in several different forms: in terms of a linearised problem, in terms of Lagrange multipliers (KKT-points), in terms of subdifferentials and normal cones, in terms of a nonsmooth penalty function, in terms of cadres with positive cadre multipliers, and in an alternance form. We describe interconnections between all these forms of necessary and sufficient o…
In mathematical economics, the used functions are, in general, considered to be quasiconcave. Moreover, they are, in many cases, separable of nature. It is known that a local maximum of a quasiconcave function is not, in general, a global maximum. In this paper we will show that this property is true when the quasiconcave function is furthermore separable. Sufficient optimality conditions for a separable quasiconcave programming will be studied in both differentiable and nondifferentiable cases. Thanks to separability condition, quasiconcave functions have nice properties in optimization probl…
A descent method for nonsmooth multiobjective optimization problems on Riemannian manifolds
该研究提出黎曼流形上非光滑多目标优化的下降法，仅需目标函数局部利普希茨连续，通过黎曼ε-次梯度凸包获下降方向结合Armijo线搜索，证明有限步收敛到帕累托最优必要条件点，数值实验验证了方法有效性。
In this paper we are concerned with generalised L 1-minimisation problems, i.e. Bolza problems involving the absolute value of the control with a control-affine dynamics. We establish sufficient conditions for the strong local optimality of extremals given by the concatenation of bang, singular and inactive (zero) arcs. The sufficiency of such conditions is proven by means of Hamiltonian methods. As a byproduct of the result, we provide an explicit invariant formula for the second variation along the singular arc.
In this article we study optimal control problems for systems that are affine with respect to some of the control variables and nonlinear in relation to the others. We consider finitely many equality and inequality constraints on the initial and final values of the state. We investigate singular optimal solutions for this class of problems, for which we obtain second order necessary and sufficient conditions for weak optimality in integral form. We also derive Goh pointwise necessary optimality conditions. We show an example to illustrate the results.
Throughout this paper, we focused our aim on the problem of optimal control under a risk-sensitive performance functional, where the system is given by a fully coupled forward-backward stochastic differential equation with jump. The risk neutral control system has been used as preliminary step, where the admissible controls are convex, and the optimal solution exists. The necessary as well as sufficient optimality conditions for risk-sensitive performance are proved. At the end of this work, we illustrate our main result by giving an example of mean-variance for risk sensitive control problem …
The Karush-Kuhn-Tucker Optimality Conditions for Multi-Objective Interval-Valued Optimization Problem on Hadamard Manifolds
本文研究了Hadamard流形上多目标区间值优化问题的KKT最优性条件，在LU和CW序下定义了Pareto最优解，并在凸性、伪凸性和广义Hukuhara差下提出KKT条件，证明了其比欧氏空间上的现有结果更具一般性。
We prove a sufficient optimality condition for non-linear optimal control problems with delays in both state and control variables. Our result requires the verification of a Hamilton-Jacobi partial differential equation and is obtained through a transformation that allow us to rewrite a delayed optimal control problem as an equivalent non-delayed one.
We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.
We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex, and the system is governed by a nonlinear backward stochastic differential equation. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality for two models. The first concerns the relaxed controls, who are measure-valued processes. The second is a particular case of the first and relates to strict control problems.
The aim of this work is to make a survey on recent sufficient optimality conditions for optimal control problems with time delays in both state and control variables. The results are obtained by transforming delayed optimal control problems into equivalent non-delayed problems. Such approach allows to use standard theorems that ensure sufficient optimality conditions for non-delayed optimal control problems. Examples are given with the purpose to illustrate the results.
In this paper studies the maximal covering location problem, assuming imprecise knowledge of all data involved. The considered problem is modeled from a fuzzy perspective producing suitable fuzzy Pareto solutions. Some properties of the fuzzy model are studied, which validate the equivalent mixed-binary linear multiobjective formulation proposed. A solution algorithm is developed, based on the augmented weighted Tchebycheff method, which produces solutions of guaranteed Pareto optimality. The effectiveness of the algorithm has been tested with a series of computational experiments, whose numer…
A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces
本文提出一种在Hilbert空间中求解非光滑多目标优化问题的下降方法，通过高效近似次微分并计算公共下降方向迭代寻找Pareto临界点，证明了收敛性，并在障碍问题最优控制中验证了效率。
Pareto optimal multi-robot motion planning
本文研究了一类多机器人协调问题，目标是使机器人以最短时间到达目标区域并避免碰撞。提出了一种新的数值算法来识别Pareto最优解，确保没有机器人可以单方面减少旅行时间而不延长其他机器人的。通过室内多机器人平台和计算机模拟实验，展示了该算法的 anytime 特性。
We are interested in the existence of Pareto solutions to the vector optimization problem $$\text{Min}_{\,\mathbb{R}^m_+} \{f(x) \,|\, x\in \mathbb{R}^n\},$$ where $f\colon\mathbb{R}^n\to \mathbb{R}^m$ is a polynomial map. By using the {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m - 1$ containing the set of Pareto values of the problem. Then we establish connections between the Palais--Smale conditions, $M$-tameness, and properness for the map $f$. Based on these results, we provide some sufficient conditions for the existence of Pareto solutions…
New Pareto Z-eigenvalue localization theorems for tensors with applications
本文为张量特征值互补问题的Pareto Z-特征值建立新定位集，并应用于TCP和TGEiCP解存在性及真空稳定性检验。
On quantitative sufficient second-order optimality conditions for elliptic optimal control problems
本文针对无Tikhonov正则化、受半线性椭圆方程约束的盒约束分布式最优控制问题，提出定量条件以保证二阶条件在状态与跟踪数据扰动下的稳定性。
The idea of iterative process optimization based on collected output measurements, or "real-time optimization" (RTO), has gained much prominence in recent decades, with many RTO algorithms being proposed, researched, and developed. While the essential goal of these schemes is to drive the process to its true optimal conditions without violating any safety-critical, or "hard", constraints, no generalized, unified approach for guaranteeing this behavior exists. In this two-part paper, we propose an implementable set of conditions that can enforce these properties for any RTO algorithm. This seco…
We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is monotone absorbing, hence maximal monotone when the function is convex.
We present a unified treatment to control problems on an arbitrary time scale by introducing the study of forward-backward optimal control problems. Necessary optimality conditions for delta-nabla isoperimetric problems are proved, and previous results in the literature obtained as particular cases. As an application of the results of the paper we give necessary and sufficient Pareto optimality conditions for delta-nabla bi-objective optimal control problems.
This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from countably many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish "no-gap" second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i. e., for which the sufficient condition only differs from the necessary one in the fact that …
Bang--bang trajectories with a double switching time: sufficient strong local optimality conditions
本文针对具有多个切换点的bang-bang极值轨迹，给出了强拓扑下局部最优的充分条件，该条件是先前特定情形条件的自然推广。
We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex and the the variable control has two components, the first being absolutely continuous and the second singular. The system is governed by a nonlinear stochastic differential equation, in which the absolutely continuous component of the control enters both the drift and the diffusion coefficients. By introducing a new approach, we establish necessary and sufficient optimality conditions for two models. The first concerns the relaxed-singular controls, who are a pair of processes whose…
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