Feedback Optimality Condition for Nonconvex Discrete Linear-Quadratic Optimal Control Problem

The paper analyzed a non-convex linear-quadratic optimization problem in a discrete dynamic system. We obtained necessary optimality condition with feedback controls which allow a descent of the functional cost. Such controls are generated by the quadratic majorant of the cost. In contrast to the discrete maximum principle, this condition does not require any convexity properties of the problem.

Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

Directional Necessary Optimality Conditions for Bilevel Programs

The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian–Fromovitz constraint qualification (MFCQ). In this paper, we show that even the first order sufficient condition for metric subregularity (which is, in general, weaker than MFCQ) fails at each feasible point of the bilevel program. We introduce the concept of a directional calmness condition and show that, under the directional calmness condition, the directional necessary optimality condition holds. Although the directional optimality condition is, in general, sharper than the nondirectional one, the directional calmness condition is, in general, weaker than the classical calmness condition and, hence, is more likely to hold. We perform the directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional calmness. An example is given to show that the directional quasi-normality condition may hold for the bilevel program.

A New Sequential Optimality Condition of Cardinality-Constrained Optimization Problem and Application

In this paper, we consider the cardinality-constrained optimization problem and propose a new sequential optimality condition for the continuous relaxation reformulation which is popular recently. It is stronger than the existing results and is still a first-order necessity condition for the cardinality constraint problem without any additional assumptions. Meanwhile, we provide a problem-tailored weaker constraint qualification, which can guarantee that new sequential conditions are Mordukhovich-type stationary points. On the other hand, we improve the theoretical results of the augmented Lagrangian algorithm. Under the same condition as the existing results, we prove that any feasible accumulation point of the iterative sequence generated by the algorithm satisfies the new sequence optimality condition. Furthermore, the algorithm can converge to the Mordukhovich-type (essentially strong) stationary point if the problem-tailored constraint qualification is satisfied.

Robust Semi-Automatic Annotation of object Data Sets with Bounding Rectangles.

Object datasets used in the construction of object detectors are typically manually annotated with horizontal or rotated bounding rectangles. The optimality of an annotation is obtained by fulfilling two conditions (i) the rectangle covers the whole object (ii) the area of ​​the rectangle is minimal. Building a large-scale object dataset requires annotators with equal manual dexterity to carry out this tedious work. When an object is horizontal, it is easy for the annotator to reach the optimal bounding box within a reasonable time. However, if the object is rotated, the annotator needs additional time to decide whether the object will be annotated with a horizontal rectangle or a rotated rectangle. Moreover, in both cases, the final decision is not based on any objective argument, and the annotation is generally not optimal. In this study, we propose a new method of annotation by rectangles, called Robust Semi-Automatic Annotation, which combines speed and robustness. Our method has two phases. The first phase consists in inviting the annotator to click on the most relevant points located on the contour of the object. The outputs of the first phase are used by an algorithm to determine a rectangle enclosing these points. To carry out the second phase, we develop an algorithm called RANGE-MBR, which determines, from the selected points on the contour of the object, a rectangle enclosing these points in a linear time. The rectangle returned by RANGE-MBR always satisfies optimality condition (i). We prove that the optimality condition (ii) is always satisfied for objects with isotropic shapes. For objects with anisotropic shapes, we study the optimality condition (ii) by simulations. We show that the rectangle returned by RANGE-MBR is quasi-optimal for the condition (ii), and that its performance increases with dilated objects, which is the case for most of the objects appearing on images collected by aerial photography.

A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier–Stokes Equations

We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Using a stochastic maximum principle, we derive a necessary optimality condition to obtain explicit formulas the optimal controls have to satisfy. Moreover, we show that the optimal controls satisfy a sufficient optimality condition. As a consequence, we are able to solve uniquely control problems constrained by the stochastic Navier–Stokes equations especially for two-dimensional as well as for three-dimensional domains.

Sequential optimality conditions for cardinality-constrained optimization problems with applications

Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.

Efficient Algorithms for Rotation Averaging Problems

The rotation averaging problem is a fundamental task in computer vision applications. It is generally very difficult to solve due to the nonconvex rotation constraints. While a sufficient optimality condition is available in the literature, there is a lack of a fast convergent algorithm to achieve stationary points. In this paper, by exploring the problem structure, we first propose a block coordinate descent (BCD)-based rotation averaging algorithm with guaranteed convergence to stationary points. Afterwards, we further propose an alternative rotation averaging algorithm by applying successive upper-bound minimization (SUM) method. The SUM-based rotation averaging algorithm can be implemented in parallel and thus is more suitable for addressing large-scale rotation averaging problems. Numerical examples verify that the proposed rotation averaging algorithms have superior convergence performance as compared to the state-of-the-art algorithm. Moreover, by checking the sufficient optimality condition, we find from extensive numerical experiments that the proposed two algorithms can achieve globally optimal solutions.

First-Order Necessary Conditions in Optimal Control

In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided conditions under which accumulation points satisfy a condition requiring a certain optimality function, used in the algorithms to generate search directions, to be nonnegative for all controls. The aim of this paper is to clarify the nature of this optimality condition, which we call the first-order minimax condition, and of a related integrated form of the condition, which, also, is implicit in past algorithm convergence analysis. We consider these conditions, separately, when a pathwise state constraint is, and is not, included in the problem formulation. When there are no pathwise state constraints, we show that the integrated first-order minimax condition is equivalent to the minimum principle and that the minimum principle (and equivalent integrated first-order minimax condition) is strictly stronger than the first-order minimax condition. For problems with state constraints, we establish that the integrated first-order minimax condition and the minimum principle are, once again, equivalent. But, in the state constrained context, it is no longer the case that the minimum principle is stronger than the first-order minimax condition, or vice versa. An example confirms the perhaps surprising fact that the first-order minimax condition is a distinct optimality condition that can provide information, for problems with state constraints, in some circumstances when the minimum principle fails to do so.