In this manuscript, we consider a control system governed by a general ordinary differential equation on a Riemannian manifold, with its endpoints satisfying some inequalities and equalities, and its control constrained to a closed convex set. We concern on an optimal control problem of this system, and obtain the second order necessary condition in the sense of convex variation (Theorem 2.2). To this end, we first obtain a second order necessary condition of an optimization problem (Theorem 4.2) via separation theorem of convex sets. Then, we derive our necessary condition by transforming the optimal control problem into an optimization problem. It is worth to point out that, our necessary condtition evolves the curvature tensor, which is trivial in Euclidean case. Moreover, even M is a Euclidean space, our result is still of interest. Actually, we give an example (Example 2.1) which shows that, when an optimal control stays at the boundary of the control set, the existing results are invalid while Theorem 2.2 works.
In this paper, we are concerned with optimal control problems evolved on Riemannian manifolds, where the initial and final states satisfy some inequality and equality type constraints, and the control set is a separable metric space. We obtain the second order necessary conditions of integral and quasi-pointwise forms, both of which work for Pontryagin type critical controls and involve the curvature tensor. Also, we apply the condition of integral form to the Bolza problem, where the initial and final states are subject to equality's type constraint.
The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed-endpoint problems for linear time-varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free-endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed-endpoin…
This paper is the first part of our series work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, both drift and diffusion terms may contain the control variable but the control region is assumed to be convex. Under some assumptions in terms of Malliavin calculus, we establish the desired necessary condition for stochastic singular optimal controls in the classical sense.
This work is concerned with an optimal control problem on a Riemannian manifold, for which two typical cases are considered. The first case is when the endpoint is free. For this case, the control set is assumed to be a separable metric space. By introducing suitable dual equations, which depend on the curvature tensor of the manifold, we establish the second order necessary and sufficient optimality conditions of integral form. In particular, when the control set is a Polish space, the second order necessary condition is reduced to a pointwise form. As a key preliminary result and also an int…
We establish necessary optimality conditions for variational problems with an action depending on the free endpoints. New transversality conditions are also obtained. The results are formulated and proved using the recent and general theory of time scales via the backward nabla differential operator.
In this paper provide sufficient and necessary conditions for the minimax equality for extended-valued $Φ$-convex functions. As an application we establish sufficient and necessary conditions for the minimax equality for convex-concave functions.
In this paper we consider three minimization problems, namely quadratic, $ρ$-convex and quadratic fractional programing problems. The quadratic problem is considered with quadratic inequality constraints with bounded continuous and discrete mixed variables. The $ρ$-convex problem is considered with $ρ$-convex inequality constraints in mixed variables. The quadratic fractional problem is studied with quadratic fractional constraints in mixed variables. For all three problems we reformulate the problem as a mathematical programming problem and apply standard Karush Kuhn Tucker necessary conditio…
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…
We provide convex necessary and sufficient conditions for the robust stability of linear positively dominated systems. In particular we show that the structured singular value is always equal to its convex upper bound for nonnegative matrices and we use this result to derive necessary and sufficient Linear Matrix Inequality (LMI) conditions for robust stability that involve only the system's static gain. We show how this approach can be applied to test the robust stability of the Foschini-Miljanic algorithm for power control in wireless networks in presence of uncertain interference.
We define Walrasian economic equilibrium in convex regions. We introduce the definition of strict-proper quasimonotone functions, and derive some necessary and sufficient conditions for the existence and uniqueness of Walrasian equilibrium vectors in convex regions.
Let us say that a convex function f\colon C\to[-\infty,\infty] on a convex set C\subseteq\R is infimum-stable if, for any sequence (f_n) of convex functions f_n\colon C\to[-\infty,\infty] converging to f pointwise, one has \inf_C f_n\to\inf_C f. A simple necessary and sufficient condition for a convex function to be infimum-stable is given. The same condition remains necessary and sufficient if one uses Moore--Smith nets (f_ν) in place of sequences (f_n). This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.
In the paper we consider the infinite horizon control problems on the interval with free right-hand endpoint. We obtain the necessary conditions of strict optimality. The method of the proof actually follows the classic paper by Halkin, and the boundary condition for infinity that we construct in our paper is a stronger variety of the Seierstad condition. The complete system of relations of the maximum principle that was obtained in the paper allows us to write the expression for the adjoint variable in the form of improper integral that depends only on the developing trajectory. S.M. Aseev, A…
The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that,…
Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions, weak and strong duality theorems. In particular, we show that the dual problem has the same structure as the primal problem, and the strong duality relation holds under three different sets of conditions.
In this paper, we provide conditions under which one can take derivatives of the solution to convex optimization problems with respect to problem data. These conditions are (roughly) that Slater's condition holds, the functions involved are twice differentiable, and that a certain Jacobian matrix is non-singular. The derivation involves applying the implicit function theorem to the necessary and sufficient KKT system for optimality.
The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in Ω} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(Ω)}$ when the supremand $f$ is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand $f$ are also investigated.
This paper focuses on second-order necessary optimality conditions for constrained optimization problems on Banach spaces. For problems in the classical setting, where the objective function is $C^2$-smooth, we show that strengthened second-order necessary optimality conditions are valid if the constraint set is generalized polyhedral convex. For problems in a new setting, where the objective function is just assumed to be $C^1$-smooth and the constraint set is generalized polyhedral convex, we establish sharp second-order necessary optimality conditions based on the Fréchet second-order subdi…
A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\cv{\cup_{k=1}^mW_k}$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient condition: the boundary $\bdS$ i…
Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convex cone $\mathcal{F}_+^{k:n}$ of all functions $f$ on an arbitrary interval $I\subseteq\mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots,n$ are nondecreasing is characterized in terms of extreme rays of the cone $\mathcal{F}_+^{k:n}$. A simple description of the convex cone dual to $\mathcal{F}_+^{k:n}$ is given. These results are useful in, and were motivated by, applications in probability. In fact, the results are obtained in a more general setting with certain generalized derivatives of $f$ of the $j$th…
CyclOT: Learning Quadratic Optimal Transport Maps via Synchronized Forward-Backward Interpolants
SLAC National Accelerator Laboratory, Stanford University（斯坦福大学SLAC国家加速器实验室）；University of California, Davis（加州大学戴维斯分校）；National Taiwan University（国立台湾大学）；University of Utah（犹他大学）
A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can choose only pure strategies, we identify a structural condition on the preference b(x,y) of agent type x for product type y -- and on the principal's costs c(y) -- which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal's problem theoretically and comp…
For the two sixth-order polynomials $a(s)$ and $b(s),$ Hurwitz stability of their convex combination is necessary and sufficient for the existence of a polynomial $c(s)$ such that $c(s)/a(s)$ and $c(s)/b(s)$ are both strictly positive real. Our reasoning method is constructive, and is insightful and helpful in solving the general robust strictly positive real synthesis problem.
We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order queries required to optimize a convex Lipschitz function subject to a memory constraint.
State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothness parameters is necessary. As a special case, we show Nesterov's accelerated cubic regularization method to be nearly tight.
