Quasi-convex Optimization Research Articles

Decomposed quasiconvex optimization with application to generalized cone problems

AbstractWe propose a gradient-based method to solve quasiconvex optimization problems through decomposed optimization and prove local superlinear convergence under mild regularity assumptions at the optimal solution. A practical implementation further provides global convergence while maintaining the fast local convergence. In numerical examples from generalized cone programming, the proposed method reduced the number of iterations to 18 to 50% compared to bisection.

On Tracking Varying Bounds When Forecasting Bounded Time Series

We consider a new framework where a continuous, though bounded, random variable has unobserved bounds that vary over time. In the context of univariate time series, we look at the bounds as parameters of the distribution of the bounded random variable. We introduce an extended log-likelihood estimation and design algorithms to track the bound through online maximum likelihood estimation. Since the resulting optimization problem is not convex, we make use of recent theoretical results on stochastic quasiconvex optimization, to eventually derive an Online Normalized Gradient Descent algorithm. We illustrate and discuss the workings of our approach based on both simulation studies and a real-world wind power forecasting problem.

Data-driven quasi-convex method for hit rate optimization of process product quality in digital twin

Hit rate is an important quantitative criterion for the process product quality prediction of the integrated industrial processes. The hit rate indicates the percentage of product quantities accepted by the downstream process within the controlled range of the product quality. The optimization model of the hit rate criterion is a non-convex intractable problem. In order to improve the hit rate of the predicted product quality, we define a hit rate optimization problem, and propose a data-driven quasi-convex approach, which converts the original problem into a set of convex feasible problems and achieves the optimal hit rate. The proposed approach combines factorial hidden Markov models, multitask elastic net and quasi-convex optimization. In order to illustrate the advantages of the proposed approach, a Monte Carlo simulation experiment is designed to verify the convex optimization property. Another experiment is carried out on two actual steel production datasets for the temperature prediction in molten iron dispatch. The results confirm that the proposed approach not only exhibits superior performance with the controlled hit rate, but also improves the hit rate by at least 41.11 % and 31.01 %, respectively, compared with the classical models on two real datasets.

Neurodynamic Flow Approach for Convex and Quasi-Convex Optimization on Riemannian Manifolds With Diagonal Metrics

Neurodynamic Flow Approach for Convex and Quasi-Convex Optimization on Riemannian Manifolds With Diagonal Metrics

Karush-Kuhn-Tucker optimality conditions for non-smooth geodesic quasi-convex optimization on Riemannian manifolds

This paper is committed to studying Karush-Kuhn-Tucker (in short, KKT) type necessary and sufficient optimality conditions for non-smooth quasi-convex (geodesic sense) optimization problems on Riemannian manifolds. Recently, Ansari et al. [Ansari QH, Babu F, Zeeshan M. Incremental quasi-subgradient method for minimizing geodesic quasi-convex function on Riemannian manifolds with applications. Numer Funct Anal Optim. 2022;42(13):1492–1521. doi: 10.1080/01630563.2021.2001823] defined the quasi-subdifferential on Riemannian manifolds and established the existence results of the quasi-subdifferential. We provide several auxiliary results for the quasi-subdifferential in the current study. We offer the KKT optimality conditions for the quasi-convex optimization problems on Riemannian manifolds with or without the Slater-constraint qualifications. To verify the suggested outcomes, we formulate numerical examples. In addition, we also provide our results in the Euclidean spaces, which are original and distinct from earlier findings in the Euclidean spaces.

Distributed control of time-delay interconnected nonlinear systems

This paper addresses the distributed control of delayed interconnected nonlinear systems with time-varying delays in both the local subsystems’ dynamics and the physical interconnections among the subsystems. The Takagi–Sugeno fuzzy model with nonlinear consequent parts (N-TS), which is capable to provide less complex representations than standard T–S fuzzy models, is considered to efficiently deal with this class of complex systems. Then, based on Lyapunov–Krasovskii stability arguments, a synthesis condition is proposed to design a distributed control law such that the origin of the closed-loop interconnected system is locally asymptotically stable together with a guaranteed set of admissible initial conditions for which the validity of the N-TS fuzzy model is ensured. Moreover, a quasi-convex optimization procedure is formulated to enlarge the set of admissible initial conditions. The effectiveness of the proposed synthesis condition is validated in two numerical examples, including an interconnected power network with seven generators.

A mixed-integer quasi-convex optimization model for joint transmission network and wind power investment problem

A mixed-integer quasi-convex optimization model for joint transmission network and wind power investment problem

On the Lower Semicontinuity of the Value Function and Existence of Solutions in Quasiconvex Optimization

On the Lower Semicontinuity of the Value Function and Existence of Solutions in Quasiconvex Optimization

Wind speed forecasting with correlation network pruning and augmentation: A two-phase deep learning method

To ensure the operational reliability of power systems, it is important for wind speed signal forecasting systems of wind turbines to be efficient, accurate and stable. This paper proposes a two-phase deep learning structure with network augmentation and pruning. By introducing the cross-correlation and quasi-convex optimization, a fractional quadratic programming problem and related convex optimization models are constructed to generate the augmented data for this proposed internal network; by pruning weakly correlated convolution channels, the redundant features of its external network are reduced. Furthermore, the closed-form solution of the convex optimization model is derived, which reduces the computational complexity considerably from O(n*log(2N)) to O(n). The proposed approach has been extensively validated using the real data of the wind farm in China. The results of the numerical experiments demonstrate that the proposed method achieves the superior performance in the training flexibility, model accuracy, stability, and interpretability.

Inexact proximal point algorithm for quasiconvex optimization problems on Hadamard manifolds

ABSTRACT In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution.

Multivariate approximation by polynomial and generalized rational functions

In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables.

Robust Structural Stability, Stability Margins, and Maximum Uncertainty Amplification for 2D Uncertain Systems via Structured Lyapunov Functions and Matrix Annihilators

This paper considers 2D uncertain systems with homogeneous or mixed dynamics, and addresses three problems: establishing robust structural stability, computing robust structural stability margins, and determining the maximum uncertainty amplification that preserves robust structural stability. Two sufficient linear matrix inequality (LMI) conditions are proposed for establishing robust structural stability obtained by introducing an equivalent closed-loop complex system and by searching for real or complex structured Lyapunov functions and matrix annihilators parameterized by the uncertainties and auxiliary quantities. Moreover, it is shown that lower bounds of the introduced robust structural stability margins and maximum uncertainty amplification can be obtained by solving quasi-convex optimization problems under some restrictions on the sought Lyapunov functions or on the set of uncertainties. Lastly, the nonconservatism of the proposed results is analyzed, showing that the proposed LMI condition based on the use of complex Lyapunov functions is not only sufficient but also necessary under some assumptions.

Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces

Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces

Multi-sensor Kalman filtering over packet-dropping networks subject to Round-Robin protocol scheduling

This paper considers the filtering problem for a class of linear cyber-physical systems (CPSs) subject to the Round-Robin protocol (RRP) scheduling, where the RRP is adopted to efficiently avoid data collisions in multi-sensor application scenarios. Unlike most of the existing results concerning the scheduling effects of the RRP under reliable communication channels, the filtering problem over packet-dropping networks is investigated. In such a framework, an optimal Kalman-type recursive filter is derived in the minimum mean square error (MMSE) sense, which is different from the suboptimal filters with bounded error covariances proposed in the previous results. Due to the protocol-induced behaviors and the unreliability of the channels, the estimator may be unstable. Thus, the stability problem of the filter is mainly discussed. It can be proved that the filter is stable when the arrival rate of the measurements exceeds a certain threshold, where the threshold can be obtained by solving a quasi-convex optimization problem. Furthermore, a sufficient condition for the existence of the steady-state error covariance is presented and can be transferred into the feasibility of a certain linear matrix inequality (LMI). Finally, a simulation example is provided to demonstrate the developed results.

The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions

In this paper, we demonstrate that a well-known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimization. Although this correspondence is not surprising, it naturally connects rational and generalized rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and, therefore, offers more theoretical and computational tools for solving this problem. The second important contribution of this paper is the extension of the linear inequality method to a broader class of Chebyshev approximation problems, where the corresponding objective functions remain quasiconvex. In this broader class of functions, the inequalities are no longer required to be linear: it is enough for each inequality to define a convex set and the computational challenge is in solving the corresponding convex feasibility problems. Therefore, we propose a more systematic and general approach for treating Chebyshev approximation problems. In particular, we are looking at the problems where the approximations are quasilinear functions with respect to their parameters that are also the decision variables in the corresponding optimization problems.

Time-Optimal Point Stabilization Control for WIP Vehicles Using Quasi-Convex Optimization and B-Spline Adaptive Interpolation Techniques

This paper addresses a time-optimal point stabilization control for inherent unstable wheeled inverted pendulum (WIP) vehicles using quasi-convex optimization and B-spline adaptive interpolation techniques. First, to handle the difficulty caused by the inherent unstable characteristic, a state feedback control law is introduced to the state-space model deduced with the kinematic coupling relationship between the longitudinal motion and tilt angle of vehicle body. Then, by system discretization, a standard quasi-convex optimization problem is formulated to plan a time-optimal trajectory with various constraints being taken into account, and the optimization problem can be solved by a bisection method combined with solving a series of convex feasibility problems. Next, to obtain the analytic expression of the discrete optimal trajectory, a B-spline interpolation algorithm with adaptive curve refinement is presented based on feature points recognition. Furthermore, as the time-optimal trajectories with analytic expressions, including displacement, velocity, and acceleration trajectories, have been obtained, a proportional-integral-derivative (PID) tracking controller is applied to control the vehicle to track the optimal trajectories and, thus, the time-optimal point stabilization control for the WIP vehicle can be realized. Finally, simulation results of a numerical example is presented to validate the feasibility and effectiveness of the proposed method.

Characterizing quasiconvexity of the pointwise infimum of a family of arbitrary translations of quasiconvex functions, with applications to sums and quasiconvex optimization

It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two statements (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of them being the quasiconvexity of the pointwise infimum of arbitrary translations of quasiconvex functions in the family; another is the convexity of the union of any two of their sublevel sets; a third one is the quasiconvexity of the sum of the quasiconvex functions, composed with arbitrary nondecreasing functions. As a by-product, any of the aforementioned characterizations, besides providing quasiconvexity of the sum, also implies the semistrict quasiconvexity of the sum if every function in the family has the same property. Three concrete applications in quasiconvex optimization are presented: First, we establish the convexity of the (Benson) proper efficient solution set to a quasiconvex vector optimization problem; second, we derive conditions that allow us to reduce a constrained optimization problem to one with a single inequality constraint, and finally, we show a class of quasiconvex minimization problems having zero duality gap.

Data-driven stability analysis of switched linear systems with Sum of Squares guarantees

Abstract We present a new data-driven method to provide probabilistic stability guarantees for black-box switched linear systems. By sampling a finite number of observations of trajectories, we construct approximate Lyapunov functions and deduce the stability of the underlying system with a user-defined confidence. The number of observations required to attain this confidence level on the guarantee is explicitly characterized. Our contribution is twofold: first, we propose a novel approach for common quadratic Lyapunov functions, relying on sensitivity analysis of a quasi-convex optimization program. By doing so, we improve a recently proposed bound. Then, we show that our new approach allows for extension of the method to Sum of Squares Lyapunov functions, providing further improvement for the technique. We demonstrate these improvements on a numerical example.

Convergence rates of subgradient methods for quasi-convex optimization problems

Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we investigate the quantitative convergence theory, including the iteration complexity and convergence rates, of various subgradient methods for solving quasi-convex optimization problems in a unified framework. In particular, we consider a sequence satisfying a general (inexact) basic inequality, and investigate the global convergence theorem and the iteration complexity when using the constant, diminishing or dynamic stepsize rules. More importantly, we establish the linear (or sublinear) convergence rates of the sequence under an additional assumption of weak sharp minima of Holderian order and upper bounded noise. These convergence theorems are applied to establish the iteration complexity and convergence rates of several subgradient methods, including the standard/inexact/conditional subgradient methods, for solving quasi-convex optimization problems under the assumptions of the Holder condition and/or the weak sharp minima of Holderian order.

Convex Optimization Over Risk-Neutral Probabilities

We consider a collection of derivatives that depend on the price of an underlying asset at expiration or maturity. The absence of arbitrage is equivalent to the existence of a risk-neutral probability distribution on the price; in particular, any risk neutral distribution can be interpreted as a certificate establishing that no arbitrage exists. We are interested in the case when there are multiple risk-neutral probabilities. We describe a number of convex optimization problems over the convex set of risk neutral price probabilities. These include computation of bounds on the cumulative distribution, VaR, CVaR, and other quantities, over the set of risk-neutral probabilities. After discretizing the underlying price, these problems become finite dimensional convex or quasiconvex optimization problems, and therefore are tractable. We illustrate our approach using real options and futures pricing data for the S&P 500 index and Bitcoin.