Interior Point Optimization Method Research Articles

Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a new type of kernel functions

Kernel functions are essential for designing and analyzing interior-point methods (IPMs). They are used to determine search directions and reduce the computational complexity of the interior point method. Currently, IPM based on kernel functions is one of the most effective methods for solving LO [1,20], second-order cone optimization (SOCO) [2], and symmetric optimization (SO) and is a very active research area in mathematicalprogramming. This paper presents a large-update primal-dual IPM for SDO based on a new bi-parameterized hyperbolic kernel function. Then we proved that the proposed large-update IPM has the same complexity bound as the best-known IPMs for solving these problems. Taking advantage of the favorable characteristics of the kernel function, we can deduce that the iteration bound for the large update method is $\mathcal{O}\left( \sqrt{n}\log n\log\dfrac{n}{\varepsilon }\right) $ when a takes a special value utilizing the favorable properties of the kernel function. These theoretical results play an essential role in the design and analysis of IPMs for CQSCO [7] and the Cartesian$\ P_{\ast }\left( \kappa \right) $-SCLCP [8]. The proximity function has never been used. In order to validate the efficacy of our algorithm and verify the effectiveness of our algorithm, examples aregiven to illustrate the applicability of our main results, and we compare our numerical results with some alternatives presented in the literature.

Certifiably optimal rotation and pose estimation based on the Cayley map

We present novel, convex relaxations for rotation and pose estimation problems that can a posteriori guarantee global optimality for practical measurement noise levels. Some such relaxations exist in the literature for specific problem setups that assume the matrix von Mises-Fisher distribution (a.k.a., matrix Langevin distribution or chordal distance) for isotropic rotational uncertainty. However, another common way to represent uncertainty for rotations and poses is to define anisotropic noise in the associated Lie algebra. Starting from a noise model based on the Cayley map, we define our estimation problems, convert them to Quadratically Constrained Quadratic Programs (QCQPs), then relax them to Semidefinite Programs (SDPs), which can be solved using standard interior-point optimization methods; global optimality follows from Lagrangian strong duality. We first show how to carry out basic rotation and pose averaging. We then turn to the more complex problem of trajectory estimation, which involves many pose variables with both individual and inter-pose measurements (or motion priors). Our contribution is to formulate SDP relaxations for all these problems based on the Cayley map (including the identification of redundant constraints) and to show them working in practical settings. We hope our results can add to the catalogue of useful estimation problems whose solutions can be a posteriori guaranteed to be globally optimal.

Online Interior-Point Methods for Time-Varying Equality-Constrained Optimization

Online Interior-Point Methods for Time-Varying Equality-Constrained Optimization

Safe Deep Reinforcement Learning for Microgrid Energy Management in Distribution Networks With Leveraged Spatial–Temporal Perception

Microgrids (MG) have recently attracted great interest as an effective solution to the challenging problem of distributed energy resources’ management in distribution networks. In this context, despite deep reinforcement learning (DRL) constitutes a well-suited model-free and data-driven methodological framework, its application to MG energy management is still challenging, driven by their limitations on environment status perception and constraint satisfaction. In this paper, the MG energy management problem is formalized as a Constrained Markov Decision Process, and is solved with the state-of-the-art interior-point policy optimization (IPO) method. In contrast to conventional DRL approaches, IPO facilitates efficient learning in multi-dimensional, continuous state and action spaces, while promising satisfaction of complex network constraints of the distribution network. The generalization capability of IPO is further enhanced through the extraction of spatial-temporal correlation features from original MG operating status, combining the strength of edge conditioned convolutional network and long short-term memory network. Case studies based on an IEEE 15-bus and 123-bus test feeders with real-world data demonstrate the superior performance of the proposed method in improving MG cost effectiveness, safeguarding the secure operation of the network and uncertainty adaptability, through performance benchmarking against model-based and DRL-based baseline methods. Finally, case studies also analyze the computational and scalability performance of proposed and baseline methods.

Dynamical model development and parameter identification for solid-state anaerobic digestion of shellfish products: Application to Mytilus edulis

A simplified AM2 model was developed to characterize mussel solid-state anaerobic digestion. This model considers two different substrates for mussels' degradation: the mussel meat and the mussel juice obtained after sanitization. This model was implemented to characterize the anaerobic degradation of Mytilus edulis species. This model was verified, implemented, and validated in 60 L batch reactors in mesophilic conditions. Two different experiments were used to calibrate kinetics using reaction invariants and an interior point optimization method. A conditioning study and a sensitivity analysis were done and had shown a better sensitivity with delayed substrate injections throughout the experiment with a factor of 10. An 88.6 % accumulation of methane yield of the BMP measurement was observed, corresponding to 57.7 % volatile removal with a minimum mass balance of 96.1 %. Additionally, the model proposed in this study was able to successfully predict the two characteristic methane yield peaks observed during solid-state anaerobic digestion.

An enhanced temporal algorithm- coupled optimized adaptive sparse principal component analysis methodology for fault diagnosis of chemical processes

Principal component analysis (PCA) is a classic fault diagnosis method widely used in chemical process data modeling. However, the limitation of PCA to handle dynamic and time-variant data has been progressively exposed with the ever-increasing development of artificial intelligence technologies and applications of automation systems in the chemical industry. In this paper, an enhanced temporal algorithm-coupled optimized adaptive sparse PCA (ETA-OASPCA) methodology is proposed with the aim to improve the conventional PCA method by introducing temporal state computation and dynamic adaptive sparsity. The ETA is constructed to extract the state transitions of the time-variant process data by introducing the “Transition” and “Hold”, acquire the region of interest by employing a Maximum Fuzzy c-means (MFCM) method, and then use the region time prediction automaton to label the possible abnormal states. The introduction of dynamic adaptive sparsity in PCA, achieved by presenting an iterative interior point optimization method to update the dynamic sparse penalty term, can impose sparsity constraints on dynamic data calculations, thus enhancing the interpretability of the PCA model. The effectiveness of the proposed method was validated through its application in the Eastman Tennessee process, showing that an average fault detection rate, false alarm rate, and latency of 99.4%, 0.37%, and 4.11 mins can be achieved, respectively.

A new search direction for full-Newton step infeasible interior-point method in linear optimization

In this work, we investigate a full Newton step infeasible interior-point method for linear optimization based on a new search direction which is obtained from an algebraic equivalent transformation of the central path system. Furthermore, we prove that the proposed method obtains an ε-optimal solution to the original problem in polynomial time.

On the implementation of a quasi-Newton interior-point method for PDE-constrained optimization using finite element discretizations

ABSTRACT We present a quasi-Newton interior-point method appropriate for optimization problems with pointwise inequality constraints in Hilbert function spaces. Among others, our methodology applies to optimization problems constrained by partial differential equations (PDEs) that are posed in a reduced-space formulation and have bounds or inequality constraints on the optimized parameter function. We first introduce the formalization of an infinite-dimensional quasi-Newton interior-point algorithm using secant BFGS updates and then proceed to derive a discretized interior-point method capable of working with a wide range of finite element discretization schemes. We also discuss and address mathematical and software interface issues that are pervasive when existing off-the-shelf PDE solvers are to be used with off-the-shelf nonlinear programming solvers. Finally, we elaborate on the numerical and parallel computing strengths and limitations of the proposed methodology on several classes of PDE-constrained problems.

Dual Certificates and Efficient Rational Sum-of-Squares Decompositions for Polynomial Optimization over Compact Sets

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. For a theoretical application, we give a short new proof of Powers's theorems on the existence of rational WSOS certificates of positive polynomials. For a computational application, we show that exact WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. We also present an algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.

Complexity analysis of primal-dual interior-point methods for linear optimization based on a new efficient Bi-parameterized kernel function with a trigonometric barrier term

In this paper we are generalizing the efficient kernel function with trigonometric barrier term given by (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Using an elegant and simple analysis and under some easy to check conditions, we explore the best complexity result for the large update primal-dual interior point methods for linear optimization. This complexity estimate improves results obtained in (X. Li and M. Zhang, Oper. Res. Lett. 43 (2015) 471–475; M.R. Peyghami and S.F. Hafshejani, Numer. Algo. 67 (2014) 33–48; M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Our comparative numerical experiments on some test problems consolidate and confirm our theoretical results according to which the new kernel function has promising applications compared to the kernel function given by (M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672). Moreover, the comparative numerical study that we have established favors our new kernel function better than other best trigonometric kernel functions (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545; M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672).

Prediction of far-field blast loads from large TNT-equivalent explosives on gabled frames

The aim of this study is to develop an empirical approach for predicting far-field blast loads from large TNT-equivalent explosives on gabled frames. For different load cases and gabled frame structures, numerical simulations were performed using a validated method. Results of the numerical simulation cases were fitted, and explicit equations for calculating the blast load parameters were derived. The polynomial functions of the peak incident overpressure and the ratio of the wavelength to the span were fitted to determine the positive phase duration time and rise time. The drag coefficients of the roof and rear wall were determined using the interior-point optimization method. The 2nd-degree polynomial logarithmic relationship between the equivalent load factor and the ratio of the wavelength to the span was fitted using the drag coefficients. The drag coefficients of the blast loads on the front wall were expressed as functions of the peak incident overpressure and the ratio of the wavelength to the span based on the numerical simulation results. The proposed approach was validated against numerical simulation cases excluded from the fitted cases.

A new corrector–predictor interior-point method for symmetric cone optimization

A new corrector–predictor interior-point method for symmetric cone optimization

A new wide-neighborhood predictor-corrector interior-point method for semidefinite optimization

A new wide-neighborhood predictor-corrector interior-point method for semidefinite optimization

A Full Nesterov-Todd Step Infeasible Interior-point Method for Symmetric Optimization in the Wider Neighborhood of the Central Path.

In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to ε-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.

An iteration predictor–corrector interior-point method with a new one-norm neighbourhood for symmetric cone optimization

In this paper, we propose a predictor–corrector interior-point method for symmetric cone optimization. The proposed algorithm is based on a new one-norm neighbourhood, which is an even wider neighbourhood than a given negative infinity neighbourhood. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We show that the algorithm has iteration complexity bound which is better than that of the usual wide neighbourhood algorithm . To our knowledge, these are the best complexity results obtained so far for the solution of symmetric cone optimization. We prove that beside the predictor steps, each corrector step also reduces the duality gap by a rate of . Finally, numerical experiments show that the proposed algorithm is efficient and reliable.

A second-order corrector wide neighborhood infeasible interior-point method for linear optimization based on a specific kernel function

In this paper, we present a second-order corrector infeasibleinterior-point method for linear optimization in a largeneighborhood of the central path. The innovation of our method is tocalculate the predictor directions using a specific kernel functioninstead of the logarithmic barrier function. We decompose thepredictor direction induced by the kernel function to two orthogonaldirections of the corresponding to the negative and positivecomponent of the right-hand side vector of the centering equation.The method then considers the new point as a linear combination ofthese directions along with a second-order corrector direction. Theconvergence analysis of the proposed method is investigated and itis proved that the complexity bound is Ο(n5/4 log e-1).

Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

The focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

A Second-Order Corrector Infeasible Interior-Point Method for Semidefinite Optimization Based on a Wide Neighborhood

This paper proposes a second-order corrector infeasible interior-point method for semidefinite optimization in a large neighborhood of the central path. Our algorithm uses the Nesterov–Todd search directions as a Newton direction. We make use of the scaled Newton directions for symmetric search directions. Based on Ai and Zhang idea, we decompose the Newton directions into two orthogonal directions corresponding to positive and negative parts of the right-hand side of the Newton equation. In such a way, we use different step lengths for each of them and the corrector step is multiplied by the square of the step length of the infeasible directions in the expression of the new iterate. Starting with a point $$(X^0, y^0, S^0)$$ in the neighborhood in terms of Frobenius norm, we show that the algorithm terminates after at most $$\mathcal {O}(n^{\frac{5}{4}}\log \varepsilon ^{-1})$$ iterations, improving by a factor $$n^{\frac{1}{4}}$$ results on these kinds of algorithms. We believe that this is the first infeasible interior-point algorithm in a large neighborhood, which uses a Newton direction decomposition and a second-order corrector step to improve the iteration complexity. The main difference of the proposed method comparing with the existing methods in literature is the strategy for generating corrector directions. Some preliminary numerical results are given to verify the efficiency of the algorithm.

A New Predictor-corrector Infeasible Interior-point Algorithm for Linear Optimization in aWide Neighborhood

In this paper, we propose a Mizuno-Todd-Ye type predictor-corrector infeasible interior-point method for linear optimization based on a wide neighborhood of the central path. According to Ai-Zhang’s original idea, we use two directions of distinct and orthogonal corresponding to the negative and positive parts of the right side vector of the centering equation of the central path. In the predictor stage, the step size along the corresponded infeasible directions to the negative part is chosen. In the corrector stage by modifying the positive directions system a full-Newton step is removed. We show that, in addition to the predictor step, our method reduces the duality gap in the corrector step and this can be a prominent feature of our method. We prove that the iteration complexity of the new algorithm is 𝒪(n log ɛ−1), which coincides with the best known complexity result for infeasible interior-point methods, where ɛ > 0 is the required precision. Due to the positive direction new system, we improve the theoretical complexity bound for this kind of infeasible interior-point method [1] by a factor of n . Numerical results are also provided to demonstrate the performance of the proposed algorithm.

An arc-search infeasible interior-point method for semidefinite optimization with the negative infinity neighborhood

We present an arc-search infeasible interior-point algorithm for semidefinite optimization using the Nesterov-Todd search directions. The algorithm is based on the negative infinity neighborhood of the central path. The algorithm searches an e-approximate solution of the problem along the ellipsoidal approximation of the entire central path. The convergence analysis of the algorithm is presented and shows that the algorithm has the iteration complexity bound $\mathcal {O}\big (n^{3/2}\log {\varepsilon }^{-1}\big )$ . Here, n is the dimension of the problem and e is the required precision. The numerical results show that our algorithm is efficient and promising.