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class="help"><a href="https://info.arxiv.org/help">Help</a> | <a href="https://arxiv.org/search/advanced">Advanced Search</a></p> </div> <div class="control"> <div class="select is-small"> <select name="searchtype" aria-label="Field to search"> <option value="all" selected="selected">All fields</option> <option value="title">Title</option> <option value="author">Author</option> <option value="abstract">Abstract</option> <option value="comments">Comments</option> <option value="journal_ref">Journal reference</option> <option value="acm_class">ACM classification</option> <option value="msc_class">MSC classification</option> <option value="report_num">Report number</option> <option value="paper_id">arXiv identifier</option> <option value="doi">DOI</option> <option value="orcid">ORCID</option> <option value="author_id">arXiv author ID</option> <option value="help">Help pages</option> <option value="full_text">Full text</option> </select> </div> </div> <input type="hidden" name="source" 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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Maia, L F"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.09927">arXiv:2407.09927</a> <span> [<a href="https://arxiv.org/pdf/2407.09927">pdf</a>, <a href="https://arxiv.org/ps/2407.09927">ps</a>, <a href="https://arxiv.org/format/2407.09927">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> An Adaptive Proximal ADMM for Nonconvex Linearly-Constrained Composite Programs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Maia%2C+L+F">Leandro Farias Maia</a>, <a href="/search/math?searchtype=author&query=Gutman%2C+D+H">David H. Gutman</a>, <a href="/search/math?searchtype=author&query=Monteiro%2C+R+D+C">Renato D. C. Monteiro</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">Gilson N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.09927v1-abstract-short" style="display: inline;"> This paper develops an adaptive Proximal Alternating Direction Method of Multipliers (P-ADMM) for solving linearly-constrained, weakly convex, composite optimization problems. This method is adaptive to all problem parameters, including smoothness and weak convexity constants. It is assumed that the smooth component of the objective is weakly convex and possibly nonseparable, while the non-smooth… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.09927v1-abstract-full').style.display = 'inline'; document.getElementById('2407.09927v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.09927v1-abstract-full" style="display: none;"> This paper develops an adaptive Proximal Alternating Direction Method of Multipliers (P-ADMM) for solving linearly-constrained, weakly convex, composite optimization problems. This method is adaptive to all problem parameters, including smoothness and weak convexity constants. It is assumed that the smooth component of the objective is weakly convex and possibly nonseparable, while the non-smooth component is convex and block-separable. The proposed method is tolerant to the inexact solution of its block proximal subproblem so it does not require that the non-smooth component has easily computable block proximal maps. Each iteration of our adaptive P-ADMM consists of two steps: (1) the sequential solution of each block proximal subproblem, and (2) adaptive tests to decide whether to perform a full Lagrange multiplier and/or penalty parameter update(s). Without any rank assumptions on the constraint matrices, it is shown that the adaptive P-ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of P-ADMMs. The two proof-of-concept numerical experiments that conclude the paper suggest our adaptive P-ADMM enjoys significant computational benefits. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.09927v1-abstract-full').style.display = 'none'; document.getElementById('2407.09927v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.08080">arXiv:2403.08080</a> <span> [<a href="https://arxiv.org/pdf/2403.08080">pdf</a>, <a href="https://arxiv.org/ps/2403.08080">ps</a>, <a href="https://arxiv.org/format/2403.08080">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> The Randomized Block Coordinate Descent Method in the H枚lder Smooth Setting </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Maia%2C+L+F">Leandro Farias Maia</a>, <a href="/search/math?searchtype=author&query=Gutman%2C+D+H">David Huckleberry Gutman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.08080v1-abstract-short" style="display: inline;"> This work provides the first convergence analysis for the Randomized Block Coordinate Descent method for minimizing a function that is both H枚lder smooth and block H枚lder smooth. Our analysis applies to objective functions that are non-convex, convex, and strongly convex. For non-convex functions, we show that the expected gradient norm reduces at an $O\left(k^{\frac纬{1+纬}}\right)$ rate, where… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.08080v1-abstract-full').style.display = 'inline'; document.getElementById('2403.08080v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.08080v1-abstract-full" style="display: none;"> This work provides the first convergence analysis for the Randomized Block Coordinate Descent method for minimizing a function that is both H枚lder smooth and block H枚lder smooth. Our analysis applies to objective functions that are non-convex, convex, and strongly convex. For non-convex functions, we show that the expected gradient norm reduces at an $O\left(k^{\frac纬{1+纬}}\right)$ rate, where $k$ is the iteration count and $纬$ is the H枚lder exponent. For convex functions, we show that the expected suboptimality gap reduces at the rate $O\left(k^{-纬}\right)$. In the strongly convex setting, we show this rate for the expected suboptimality gap improves to $O\left(k^{-\frac{2纬}{1-纬}}\right)$ when $纬>1$ and to a linear rate when $纬=1$. Notably, these new convergence rates coincide with those furnished in the existing literature for the Lipschitz smooth setting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.08080v1-abstract-full').style.display = 'none'; document.getElementById('2403.08080v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a 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