Alpelisib and Paclitaxel in PIK3CA-altered Gastric Cancer

Inclusion Criteria: 1. Subject has signed the Informed Consent Form (ICF) prior to any screening procedures being performed 2. Age ≥ 20 years old of male and female 3. At each phase of the trial, subjects who meet the following requirements in each phase will be enrolled. * Phase IB: Subjects with a histologically-confirmed, advanced/recurrent solid tumor who have progressed on standard therapy or whose disease does not have established standard therapy. * Phase II: Subjects with histologically confirmed locally advanced or metastatic gastric cancer that have progressed after treatment with first-line fluoropyrimidine-based chemotherapy (Tissue samples of gastric cancer must contain PIK3CA gene alterations (e.g. single nucleotide variants, small indels, amplifications, structural variations, etc.) identified by central or local next generation sequencing (NGS). If the subject received adjuvant chemotherapy after curative gastric resection and lymph node dissection, the adjuvant chemotherapy is considered to be the first-line palliative chemotherapy if the disease recurred during adjuvant chemotherapy or within 6 months after the completion of adjuvant chemotherapy. 4. Phase IB: Patient has evaluable disease as per RECIST 1.1. (Measurable lesions are not mandatory for study inclusion.) Phase II: Patient has at least one measurable lesion as per RECIST 1.1. 5. ECOG performance status 0-1 6. Patient has adequate bone marrow and organ function as defined by the following laboratory values: * Absolute neutrophil count (ANC) ≥ 1.5 x 109/L * Hemoglobin ≥ 9.0 g/dL * Platelet ≥ 100 x 109/L * Serum creatinine ≤ ULN (upper limit of normal) or serum creatinine clearance ≥50 mL/min (by Cockcroft-Gault formula, or 24h urine collection) * Total bilirubin: ≤ 1.5 × ULN Subjects with a bile duct obstruction will be eligible if they meet the criteria after appropriate bile drainage; Patients with Gilbert syndrome should also be included after confirming that the total bilirubin level is ≤ 1.5 x ULN in a follow-up screening test. * Phase Ib: Alanine aminotransferase (AST) and aspartate aminotransferase (ALT) ≤ 3 x ULN (regardless of liver metastases) * Phase II: AST and ALT ≤ 3 x ULN if liver metastases are absent, or AST and ALT ≤ 5 x ULN if liver metastases are present. 7. The subject is able to swallow and retain oral medication 8. Serum β-HCG test negative within 14 days before the first administration of the study treatment (women of childbearing potential only). 9. Requirement for contraception must be observed by the subject. Exclusion Criteria: 1. Patient has received previous treatment with a PI3K or AKT inhibitor. (Note: prior mTOR inhibitor treatment is allowed.) 2. Patient has a known or suspicious hypersensitivity to paclitaxel or other products containing Cremophor. 3. Any cytotoxic chemotherapy from a previous treatment regimen within 14 days. If the subject received an investigational drug from another clinical trial, the subject can be enrolled after 2 weeks of last administration and more than 5 x half-life of the investigational drug. If monoclonal antibody therapy was given, the subject can be enrolled after four weeks after the last does. 4. Active central nervous system (CNS) lesions (i.e., those with radiologically unstable or symptomatic brain lesions). For those who receive radiation or surgical treatment, the subject can be enrolled if the subject is maintained without steroid therapy and the evidence of CNS disease progression for more than 4 weeks. However, patients with leptomeningeal metastases are excluded. 5. Patient has not recovered to ≤ grade 1 (except alopecia) from related adverse effects of any prior antineoplastic therapy 6. Radiotherapy with a wide field (more than 30% of the bone marrow) of radiation within 4 weeks or radiotherapy with a limited field of radiation for palliation within 2 weeks of the first dose of study treatment. 7. Patient who has undergone major surgery ≤ 4 weeks prior to starting study treatment or who has not recovered from adverse effects of such procedure. 8. Patient has a clinically significant cardiac disease or impaired cardiac function, such as: * Acute coronary syndrome within the 6 months prior to the initiation of study drug (including myocardial infarction or unstable angina, Coronary Artery Bypass Graft surgery, percutaneous coronary intervention and stenting) * Heart failure ≥ grade 2 by New York Heart Association (NYHA) functional classification or that requires treatment * Ejection fraction (EF) \<50% on multi-gated acquisition (MUGA) scan or echocardiography examination. MUGA scan or echocardiography is not required as a screening test if there is no current suspicious symptom and past history of heart failure. * Persistent uncontrolled hypertension as defined by: systolic \>180 mmHg or diastolic \>100 mmHg despite medical treatment * Current or past history of clinically significant cardiac arrhythmia, atrial fibrillation, and/or conduction abnormality including complete AV block, long QT syndrome, congenital long QT syndrome, or QTcF \>470 msec at screening (if the average QTcF value \> 470 msec by measuring 3 times consecutively in total). * Any risk factors that prolong QTc or increase the probability of arrhythmia, including medication (e.g. heart failure, hypokalemia, congenital long QT syndrome, history of Torsades de Pointes) 9. If the subject was diagnosed with diabetes (irrespective of treatment or symptom) or if the subject has ① Korean Diabetes Prediction Score (Appendix A) more than 7 plus impaired glucose tolerance (with blood glucose of 140-199 mg/dL after 2-hour oral glucose tolerance test (75g)), ② previous history of gestational diabetes, or ③ steroid-induced diabetes. 10. Patients with impaired gastrointestinal (GI) function or GI disease that may significantly alter the absorption of oral BYL719 (e.g. untreated peptic ulcer disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, or wide small bowel resection). 11. Patient has a known positive serology for human immunodeficiency virus (HIV), active Hepatitis B, and/or active Hepatitis C infection. Hepatitis B carriers may be enrolled if prophylactic use of an antiviral agent with minimal interaction with CYP3A4 is administered to inhibit HBV activation (e.g., entecavir, adefovir) 12. Concomitant medication of strong or moderate inducers or inhibitors of CYP3A4 (Table 11) before the first dose of study treatment (In this case, if the drug is stopped for 1 week or more (according to Table 11) and changed to another drug that does not affect CYP3A4, then the subject can be enrolled.) 13. History of other primary cancer. Exceptions are as follows: * Adequately treated non-melanoma skin cancer (basal cell or squamous cell carcinoma), curatively treated in situ cancer of the cervix or stage I bladder cancer, completely resected thyroid cancer without distant metastasis in which all treatment has been completed (Appropriate wound healing is required prior to clinical trial enrollment) * Other curatively treated solid tumors except for gastric cancer with no evidence of disease recurrence at least 24 months before participating in this trial 14. History of allogeneic bone marrow transplantation or organ transplantation 15. As judged by the Investigator, all other symptoms and associated disease for which the investigator determined that participation in this study is contraindicated (e.g. Infection/inflammation; severe liver dysfunction; bilateral diffuse interstitial lung disease; uncontrolled renal disease; unstable heart and lung disease; hemorrhagic disease; intestinal obstruction; unable to swallow oral pills; social and psychological problems, etc.) 16. Medical, psychiatric, cognitive, or other conditions that may interfere with the ability of the subject to understand the subject information, provide the informed consent, follow the protocol process, or complete the clinical trial